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CcLt. 



^ 



THE 



THEOBY OF SOUND 



THE 



THEORY OF SOUND 



BY 



JOHN WILLIAM STRUTT, BARON RAYLEIGH, Sc.D., F.R.S. 

HONORARY FELLOW OF TRINITY COLLEGE, CAMBRIDGE. 



IN TWO VOLUMES 

VOLUME I. 

SECOND EDITION REVISED AND ENLARGED 



Contron : 

MACMILLAN AND CO. 

AND NEW YORK 
1894 

[All Right f( rexerved,] 



First Edition prinUd 1877. 
Second Edition revised and enlarged 1894. 



GAMBBIDOS: PBINTBD BY C. J. CLAT, M.A. & S0V8, 
AT THE UKIVBBSITT PBB88. 



PREFACE. 

IN the work, of which the present volume is an instalment, 
my endeavour has been to lay before the reader a connected 
exposition of the theory of sound, which should include the 
more important of the advances made in modem times by Mathe- 
maticians and Physicists. The importance of the object which 
I have had in view will not, I think, be disputed by those com- 
petent to judge. At the present time many of the most valuable 
contributions to science are to be found only in scattered 
periodicals and transactions of societies, published in various 
parts of the world and in several languages, and are often 
practically inaccessible to those who do not happen to live in 
the neighbourhood of large public libraries. In such a state of 
things the mechanical impediments to study entail an amount 
of unremunerative labour and consequent hindrance to the 
advancement of science which it would be difficult to over- 
estimate. 

Since the well-known Article on Sound in the Encyclopcedia 
MetropolUana, by Sir John Herschel (1845), no complete work 
has been published in which the subject is treated mathemati- 
cally. By the premature death of Prof. Donkin the scientific 
world was deprived of one whose mathematical attainments in 
combination with a practical knowledge of music qualified him 
in a special manner to write on Sound. The first part of his 
AcoiLStics (1870), though little more than a fragment, is sufficient 
to shew that my labours would have been unnecessary had Prof. 
Donkin lived to complete his work. 



VI PREFACE. 

In the choice of topics to be dealt with in a work on Sound, 
I have for the most part followed the example of my predecessors. 
To a great extent the theory of Sound, as commonly understood, 
covers the same ground as the theory of Vibrations in general ; 
but, unless some limitation were admitted, the consideration of 
such subjects as the Tides, not to speak of Optics, would have 
to be included. As a general rule we shall confine ourselves to 
those classes of vibrations for which our ears afford a ready 
made and wonderfully sensitive instrument of investigation. 
Without ears we should hardly care much more about vibrations 
than without eyes we should care about light. 

The present volume includes chapters on the vibrations of 
sjrstems in general, in which, I hope, will be recognised some 
novelty of treatment and results, followed by a more detailed 
consideration of special systems, such as stretched strings, bars, 
membranes, and plates. The second volume, of which a con- 
siderable portion is already written, will commence with a^'rial 
vibrations. 

My best thanks are due to Mr H. M. Taylor of Trinity College, 
Cambridge, who has been good enough to read the proofs. By 
his kind assistance several errors and obscurities have been 
eliminated, and the volume generally has been rendered less im- 
perfect than it would otherwise have been. 

Any corrections, or suggestions for improvements, with which 
my readers may favour me will be highly appreciated. 

Teblimo Place, Witham, 
April, 1877. 



IN this second edition all corrections of importance are noted, 
and new matter appears either as fresh sections, e.g. § 32 a, 
or enclosed in square brackets [ ]. Two new chapters X A, X B 
are interpolated, devoted to Curved Plates or Shells, and to 
Electrical Vibrations. Much of the additional matter relates to 
the more difficult parts of the subject and will be passed over 
by the reader on a first perusal. 



PREFACE. Vll 

In the mathematical investigations I have usually employed 
such methods as present themselves naturally to a physicist. 
The pure mathematician will complain, and (it must be confessed) 
sometimes with justice, of deficient rigour. But to this question 
there are two sides. For, however important it may be to 
maintain a uniformly high standard in pure mathematics, the 
physicist may occasionally do well to rest content with argu- 
ments which are fairly satisfactory and conclusive from his point 
of view. To his mind, exercised in a diflFerent order of ideas, 
the more severe procedure of the pure mathematician may appear 
not more but less demonstrative. And further, in many cases 
of difficulty to insist upon the highest standard would mean 
the exclusion of the subject altogether in view of the space 
that would be required. 

In the first edition much stress was laid upon the establish- 
ment of general theorems by means of Lagi'ange's method, and 
I am more than ever impressed with the advantages of this 
procedure. It not unfrequently happens that a theorem can be 
thus demonstrated in all its generality with less mathematical 
apparatus than is required for dealing with particular cases by 
special methods. 

During the revision of the proof-sheets I have again had the 
very great advantage of the cooperation of Mr H. M. Taylor, 
until he was unfortunately compelled to desist. To him and 
to several other friends my thanks are due for valuable sug- 
gestions. 



Jtdy, 1894. 



CONTENTS. 



PAOB 



CHAPTER I. 

§§ 1 — 27 INTRODUCTION . . . . 1 

Sonnd due to Vibrations. Finite velocity of Propagation. Velocity inde- 
pendent of Pitch. Begnault's experiments. Sonnd propagated in water. 
Wheatstone^s experiment. Enfeeblement of Sound by distance. Notes 
and Noises. Musical notes due to periodic vibrations. Siren of Cagniard 
de la Tour. Pitch dependent upon Period. Belationship between 
mnsical notes. The same ratio of periods corresponds to the same 
interval in all parts of the scale. Harmonic scales. Diatonic scales. 
Absolute Pitch. Necessity of Temperament. Equal Temperament. 
Table of Frequencies. Analysis of Notes. Notes and Tones. Quality 
dependent upon harmonic overtones. Kesolution of Notes by ear un- 
certain. Simple tones correspond to simple pendulous vibrations. 



CHAPTER II. 
§§ 28—42 a HARMONIC motions .... 19 

Composition of harmonic motions of like period. Harmonic Curve. Com- 
position of two vibrations of nearly equal period. Beats. Fourier's 
Theorem. [Beats of approximate consonances.] Vibrations in perpen- 
dicular directions. Lissajous' Cylinder. Lissajous* Figures. Black- 
burn's pendulum. Ealeidophone. Optical methods of composition 
and analysis. The vibration microscope. Intermittent Illumination. 
[Resultant of a large number of vibrations whose phases are accidentally 
distributed.] 



X CONTENTS. 

CHAPTER III. 

PAGE 

§§43 — 68 d SYSTEMS HAVING ONE DEGREE OF FREEDOM . 43 

Independence of amplitude and period. Frictional force proportional to 
velocity. Forced vibrations. Principle of Superposition. Beats due 
to superposition of forced and free vibrations. Various degrees of 
damping. String with Load. Method of Dimensions. Ideal Tuning- 
fork. Forks give nearly pure tones. Forks as standards of pitch. 
[Dependence upon temperature. Slow versus quick beats.] Scheibler*s 
methods of tuning. Scheibler's Tonometers. Compound Pendulum. 
Forks driven by electro-magnetism. [Phonic wheel.] Fork Interrupter. 
Resonance. [Intermittent vibrations.] General solution for one degree 
of freedom. [Instability.] Terms of the second order give rise to 
derived tones. [Maintenance. Methods of determining absolute pitch.] 

CHAPTER IV. 

§§ 69 — 95 VIBRATING SYSTEMS IN GENERAL . .91 

Generalized co-ordinates. Expression for potential energy. Statical theo- 
rems. Initial motions. Expression for kinetic energy. Reciprocal 
theorem. Thomson's [Kelvin's] theorem. Lagrange's equations. The 
dissipation function. Coexistence of small motions. Free vibrations 
without friction. Normal co-ordinates. The free periods fulfil a 
stationary condition. An accession of inertia increases the free periods. 
A rela;iation of spring increases the free periods. The greatest free 
' period is an absolute maximum. Hypothetical types of vibration. 
EiLample from string. Approximately simple systems. String of 
variable density. Normal functions. Conjugate property. [Introduc- 
tion of one constraint. Several constraints.] Determination of con- 
stants to suit arbitrary initial conditions. Stokes' theorem. 

CHAPTER V. 

§§96 — 117 VIBRATING SYSTEMS IN GENERAL. . . 130 

Cases in which the three functions T, F, V are simultaneously reducible to 
sums of squares. Generalization of Young's theorem on the nodal 
points of strings. Equilibrium theory. Systems started from rest as 
deflected by a force applied at one point. Systems started from the 
equilibrium configuration by an impulse applied at one point. Systems 
started from rest as deflected by a force uniformly distributed. Influ- 
ence of small frictional forces on the vibrations of a system. Solution 
of the general equations for free vibrations. [Routh's theorems. In- 
stability.] Impressed Forces. Principle of the persistence of periods. 
Inexorable motions. Reciprocal Theorem. Application to free vibrations. 
Statement of reciprocal theorem for harmonic forces. Applications. 
Extension to cases in which the constitution of the system is a function 
of the period. [Reaction at driving point.] Equations for two degrees 
of freedom. Roots of determinantal equation. Intermittent vibrations. 
March of periods as inertia is gradually increased. Reaction of a 
dependent system. 



CONTENTS. XI 

CHAPTER VI. 

PAGE 

§§118 — 148 c TRANSVERSE VIBRATIONS OF STRINGS . . 170 

Liaw of extension of a string. Transverse vibrations. Solution of the pro- 
blem for a string whose mass is concentrated in equidistant points. 
Derivation of solution for continuous string. Partial differential equa- 
tion. Expressions for V and T. Most general form of simple harmonic 
motion. Strings with fixed extremities. General motion of a string 
periodic. Mersenne's Laws. Sonometer. Normal modes of vibration. 
Determination of constants to suit arbitrary initial circumstances. Case 
of plucked string. Expressions for T and V in terms of normal co-ordi- 
nates. Normal equations of motion. String excited by plucking. 
Young's theorem. String excited by an impulse. Problem of piano- 
forte string. Friction proportional to velocity. Comparison with equi- 
librium theory. Periodic force applied at one point. Modifications due 
to yielding of the extremities. Proof of Fourier's theorem. Effects 
of a finite load. Correction for rigidity. Problem of violin string. 
Strings stretched on curved surfaces. Solution for the case of the 
sphere. Correction for irregularities of density. [Arbitrary displace- 
ment of every period.] Theorems of Sturm and Liouville for a string 
of variable density. [Density proportional to x-^. Nodes of forced vibra- 
tions.] Propagation of waves along an unlimited string. Positive and 
negative waves. Stationary Vibrations. Reflection at a fixed point. 
Deduction of solution for finite string. Grai)hical method. Progressive 
wave with friction. [Reflection at a junction of two strings. Gradual 
transition. Effect of imperfect flexibility.] 

CHAPTER VII. 

§§149—159 LONGITUDINAL AND TORSIONAL VIBRATIONS 

OF BARS 242 

Classification of the vibrations of Bars. Differential equation for longi- 
tudinal vibrations. Numerical valtfes of the constants for steel. Solu- 
tion for a bar free at both ends. Deduction of solution for a bar with 
one end free, and one fixed. Both ends fixed. Influence of small load. 
Solution of problem for bar with largo load attached. [Reflection at a 
junction.] Correction for lateral motion. Savart's *'8on rauque." 
Differential equation for torsional vibrations. Comparison of velocities 
of longitudinal and torsional waves. 

CHAPTER VIIL 

§§ 160 — 192 a LATERAL VIBRATIONS OF BARS . . . 255 

Potential energy of bending. Expression for kinetic energy. Derivation 
of differential equation. Terminal conditions. General solution for 
a harmonic vibration. Conjugate property of the normal functions. 
Values of integrated squares. Expression of T in terms of normal 



Xll CONTENTS. 

PAGE 

oo-ordinates. Normal equations of motion. Determination of constants 
to suit initial conditions. Case of rod started by a blow. Bod started 
from rest as deflected by a lateral force. In certain cases the series of 
normal functions ceases to converge. Form of the normal fonctions for 
a free-free bar. Laws of dependence of frequency on length and thick- 
ness. [Numerical formulas for tuning-forks.] Case when both ends are 
clamped. Normal functions for a clamped-free bar. Calculation of 
periods. Comparisons of pitch. Discussion of the gravest mode of 
vibration of a free-free bar. Three nodes. Four nodes. Gravest mode 
for clamped-free bar. Position of nodes. Supported bar. Calculation 
of period for clamped-free bar from hypothetical type. Solution of 
problem for a bar with a loaded end. Effect of additions to a bar. 
Influence of irregularities of density. Correction for rotatory inertia. 
Boots of functions derived linearly from normal functions. Formation 
of equation of motion when there is permanent tension. Special ter- 
minal conditions. Besultant of two trains of waves of nearly equal 
period. Fourier's solution of problem for infinite bar. [Circular Bing.] 



CHAPTER IX. 

§§193 — 213 a VIBRATIONS OF MEMBRANES . . . 306 

Tension of a membrane. Equation of motion. Fixed rectangular bound- 
ary. Expression for V and T in terms of normal co-ordinates. Normal 
equations of vibration. Examples of impressed forces. Frequency for 
an elongated rectangle depends mainly on the shorter side. Cases in 
which different modes of vibration have the same period. Derived 
modes thence arising. Effect of slight irregularities. An irregularity 
may remove indeterminateness of normal modes. Solutions applicable 
to a triangle. Expression of the general differential equation by polar 
co-ordinates. Of the two functions, which occur in the solution, one is 
excluded by the condition at the pole. Expressions for Bessel's func- 
tions. FormulsB relating thereto. Table of the first two functions. 
Fixed circular boundary. Conjugate property of the normal functions 
without restriction of boundary. Values of integrated squares. Ex- 
pressions for T and V in terms of normal functions. Normal equations 
of vibration for circular membrane. Special case of free vibrations. 
Vibrations due to a harmonic force uniformly distributed. [Force 
applied locally at the centre.] Pitches of the various simple tones. 
Table of the roots of Bessel's functions. Nodal Figures. Circular 
membrane with one radius fixed. Bessel's functions of fractional order. 
Effect of small load. Vibrations of a membrane whose boundary is 
approximately circular. In many cases the pitch of a membrane may 
be calculated from the area alone. Of all membranes of equal area that 
of circular form has the gravest pitch. Pitch of a membrane whose 
boundary is an ellipse of small eccentricity. Method of obtaining limits 
in cases that cannot be dealt with rigorously. Comparison of fre- 
quencies in various cases of membranes of equal area. History of the 
problem. Bourget's experimental investigations. [Kettle-drums. Nodal 
curves of forced vibrations.] 



CONTENTS. XUl 



CHAPTER X. 

PAGE 

§§ 214 — 235 a vibrations of plates . . . 352 

Potential Energy of Bending. Transformation of dT. Saperfioial differ- 
ential equation. Boundary conditions. Conjugate property of normal 
functions. Transformation to polar co-ordinates. Form of general 
solution continuous through pole. Equations determining the periods 
for a free circular plate. Kirchhoff's calculations. Comparison with 
observation. Badii of nodal circles. Generalization of solution. Ir- 
regularities give rise to beats. [Oscillation of nodes.] Case of clamped, 
or supported, edge. [Telephone plate.] DLsturbance of Chladni's 
figures. [Movements of sand.] History of problem. Mathieu's criti- 
cisms. Rectangular plate with supported edge. Rectangular plate with 
free edge. Boundary conditions. One special case (a(=0) is amenable 
to mathematical treatment. Investigation of nodal figures. Wheat- 
stone's application of the method of superposition. Comparison of 
Wheatstone*s figures with those really applicable to a plate in the case 
/A=0. Gravest mode of a square plate. Calculation of period on hypo- 
thetical type. Nodal figures inferred from considerations of symmetry. 
Hexagon. Comparison between circle and square. Law connecting 
pitch and thickness. In the case of a clamped edge any contraction of 
the boundary raises the pitch. No gravest form for a free plate of 
given area. In similar plates the period is as the linear dimension. 
Wheatstone's experiments on wooden plates. Koonig's experiments. 
Vibrations of cylinder, or ring. Motion tangential as well as normal. 
Relation between tangential and normal motions. Expressions for 
kinetic and potential energies. Equations of vibration. Frequencies 
of tones. Comparison with Chladni. [Fenkner's observations.] Tan- 
gential friction excites tangential motion. Experimental verification. 
Beats due to irregularities. [Examples of glass bells. Church bells.] 



CHAPTER Xa. 
§ 235 h — 235 h CURVED plates or shells . . 395 

[Extensional Vibrations. Frequency independent of thickness. luexten- 
sional or flexural vibrations. Frequency proportional to thickness. 
General conditions of inextension. Surface of second degree. Applica- 
tion to sphere. Principal extensions of cylindrical surface. Potential 
energy. Frequencies of extensional vibrations. Plane plate. Other 
particular cases of cylinder. Potential energy of bending. Sphere. 
Plane plate. Potential energy for cylindrical shell. Statical problems. 
Frequency of flexural vibrations of cylindrical shell. Extensional 
vibrations of spherical shelL Flexural vibrations of spherical shell. 
Normal modes. Potential energy. Kinetic energy. Frequencies in case 
of hemisphere. Saucer of 120^. References.] 



XIV CONTENTS. 



CHAPTER Xb. 



PAQB 



§ 235 i — 235 y electrical vibrations . . 433 

[Calculation of periods. Forced vibratioDs. Insertion of a ley den equivalent 
to a negative inductance. Initial currents in a secondary circuit. In- 
versely as the number of windings. Reaction of secondary circuit. 
Train of circuits. Initial currents alternately opposite in sign. Per- 
sistences. Resistance and inductance of two conductors in parallel. 
Extreme values of frequency. Contiguous wires. Several conductors in 
parallel Induction balance. Theory for simple harmonic currents. Two 
conditions necessary for balance. Wheatstone's bridge. Generalized 
resistance. Current in bridge. Approximate balance. Hughes' ar- 
rangement. Interrupters. Inductometers. Symmetrical arrangement. 
. Electromagnetic screening. Cylindrical conducting core. Time-con- 
stant of free currents. Induced electrical vibrations. Reaction upon 
primary circuit. Induced currents in a wire. Maxwell's formulas. 
Impedance. Kelvin's theory of cables. Heaviside's generalization. 
Attenuation and distortion of signals. Bell's telephone. Push and 
pull theory. Experiment upon bipolar telephone. Minimum current 
audible. Microphone.] 



CHAPTER I. 



INTRODUCTION. 

1. The sensation of sound is a thing sui generis, not com- 
parable with any of our other sensations. No one can express 
the relation between a sound and a colour or a smell. Directly 
or indirectly, all questions connected with this subject must 
come for decision to the ear, as the organ of hearing; and 
from it there can be no appeal. But we are not therefore to 
infer that all acoustical investigations are conducted with the 
unassisted ear. When once we have discovered the physical 
phenomena which constitute the foundation of sound, our ex- 
plorations are in great measure transferred to another field lying 
within the dominion of the principles of Mechanics. Important 
laws are in this way arrived at, to which the sensations of the ear 
cannot but conform. 

2. Very cursory observation often suffices to shew that 
sounding bodies are in a state of vibration, and that the phe- 
nomena of sound and vibration are closely connected. When a 
vibrating bell or string is touched by the finger, the sound ceases 
at the same moment that the vibration is damped. But, in order 
to affect the sense of hearing, it is not enough to have a vibrating 
instrument ; there must also be an uninterrupted communication 
between the instrument and the ear. A bell rung in vacuo, with 
proper precautions to prevent the communication of motion, 
remains inaudible. In the air of the atmosphere, however, 
sounds have a universal vehicle, capable of conveying them 
without break from the most variously constituted sources to 
the recesses of the ear. 

3. The passage of sound is not instantaneous. When a gun 
is fired at a distance, a very perceptible interval separates the 

" B. 1 



2 INTRODUCTION. [3. 

report from the flash. This represents the time occupied by 
sound in travelling from the gun to the observer, the retardation 
of the flash due to the finite velocity of light being altogether 
negligible. The first accurate experiments were made by some 
members of the French Academy, in 1738. Cannons were fired, 
and the retardation of the reports at different distances observed. 
The principal precaution necessary is to reverse alternately the 
direction along which the sound travels, in order to eliminate the 
influence of the motion of the air in mass. Down the wind, for 
instance, sound travels relatively to the earth faster than its 
proper rate, for the velocity of the wind is added to that proper 
to the propagation of sound in still air. For still dry air at a 
temperature of O^C, the French observers found a velocity of 337 
metres per second. Observations of the same character were 
made by Arago and others in 1822 ; by the Dutch physicists Moll, 
van Beek and Kuytenbrouwer at Amsterdam; by Bravais and 
Martins between the top of the Faulhom cmd a station below; 
and by others. The general result has been to give a somewhat 
lower value for the velocity of sound — about 332 metres per 
second. The effect of alteration of temperature and pressure on 
the propagation of sound will be best considered in connection with 
the mechanical theory. 

4. It is a direct consequence of observation, that within wide 
limits, the velocity of sound is independent, or at least very nearly 
independent, of its intensity, and also of its pitch. Were this 
otherwise, a quick piece of music would be heard at a little 
distance hopelessly confused and discordant. But when the dis- 
turbances are very violent and abrupt, so that the alterations of 
density concerned are comparable with the whole density of the 
air, the simplicity of this law may be departed from. 

6. An elaborate series of experiments on the propagation of 
sound in long tubes (water-pipes) has been made by Regnault*. 
He adopted an automatic arrangement similar in principle to that 
used for measuring the speed of projectiles. At the moment when 
a pistol is fired at one end of the tube a wire conveying an electric 
current is ruptured by the shock. This causes the withdrawal of a 
tracing point which was previously marking a line on a revolving 
drum. At the further end of the pipe is a stretched membrane so 
arranged that when on the arrival of the sound it yields to the 

1 Mimoiret de VAcadimie de France^ t. xxxvn. 



5.] VELOCITY OF SOUND. 3 

impulse, the circuit, which was ruptured during the passage of the 
sound, is recompleted. At the same moment the tracing point 
falls back on the drum. The blank space left unmarked corre- 
sponds to the time occupied by the sound in making the journey, 
and, when the motion of the drum is known, gives the means of 
detei-mining it. The length of the journey between the first wire 
and the membrane is found by direct measurement. In these 
experiments the velocity of sound appeared to be not quite inde- 
pendent of the diameter of the pipe, which varied from (T'lOS 
to I^'IOO. The discrepancy is perhaps due to friction, whose 
influence would be greater in smaller pipes. 

6. Although, in practice, air is usually the vehicle of sound, 
other gases, liquids and solids are equally capable of conveying 
it. In most cases, however, the means of making a direct measure- 
ment of the velocity of sound are wanting, and we are not yet in 
a position to consider the indirect methods. But in the case of 
water the same difficulty does not occur. In the year 1826, 
Colladon and Sturm investigated the propagation of sound in the 
Lake of Geneva. The striking of a bell at one station was 
simultaneous with a flash of gunpowder. The observer at a 
second station measured the interval between the flash and the 
arrival of the sound, applying his ear to a tube carried beneath 
the surface. At a temperature of 8®C., the velocity of sound in 
water was thus found to be 1435 metres per second. 

7. The conveyance of sound by solids may be illustrated by a 
pretty experiment due to Wheatstone. One end of a metallic wire 
is connected with the sound-board of a pianoforte, and the other 
taken through the partitions or floors into another part of the 
building, where naturally nothing would be audible. If a reso- 
nance-board (such as a violin) be now placed in contact with the 
wire, a tune played on the piano is easily heard, and the sound 
seems to emanate from the resonance-board. [Mechanical tele- 
phones upon this principle have been introduced into practical 
use for the conveyance of speech.] 

8. In an open space the intensity of sound falls off* with great 
rapidity as the distance from the source increases. The same 
amount of motion has to do duty over surfaces ever increasing as 
the squares of the distance. Anything that confines the sound 
will tend to diminish the falling off" of intensity. Thus over the 
flat sur£Btce of still water, a sound carries further than over broken 

1—2 



4 INTRODUCTION. [8. 

ground; the comer between a smooth pavement and a vertical 
wall is still better ; but the most effective of all is a tube-like 
enclosure, which prevents spreading altogether. The use of 
speaking tubes to facilitate communication between the different 
parts of a building is well known. If it were not for certain effects 
(frictional and other) due to the sides of the tube, sound might 
be thus conveyed with little loss to very great distances. 

9. Before proceeding further we must consider a distinction, 
which is of great importance, though not free from difficulty. 
Sounds may be classed as musical and unmusical; the former 
for convenience may be called notes and the latter noises. The 
extreme cases will raise no dispute; every one recognises the 
difference between the note of a pianoforte and the creaking of a 
shoe. But it is not so easy to draw the line of separation. In the 
first place few notes are &ee from all unmusical accompaniment. 
With organ pipes especially, the hissing of the wind as it escapes 
at the mouth may be heard beside the proper note of the pipe. 
And, secondly, many noises so far partake of a musical character 
as to have a definite pitch. This is more easily recognised in a 
sequence, giving, for example, the common chord, than by continued 
attention to an individual instance. The experiment may be made 
by drawing corks fix)m bottles, previously tuned by pouring water 
into them, or by throwing down on a table sticks of wood of suitable 
dimensions. But, although noises are sometimes not entirely 
unmusical, and notes are usually not quite free from noise, there is 
no difficulty in recognising which of the two is the simpler pheno- 
menon. There is a certain smoothness and continuity about the 
musical note. Moreover by sounding together a variety of notes — 
for example, by striking simultaneously a number of cousecutive 
keys on a pianoforte — we obtain an approximation to a noise; 
while no combination of noises could ever blend into a musical 
note. 

10. We are thus led to give our attention, in the first instance^ 
mainly to musical sounds. These arrange themselves naturally 
in a certain order according to pitch — a quality which all can 
appreciate to some extent. Trained ears can recognise an enormous 
number of gradations — more than a thousand, probably, within 
the compass of the human voice. These gradations of pitch are 
not, like the degrees of a thermometric scale, without special 
mutual relations. Taking any given note as a starting point. 



10.] PITCH. 5 

musiciaDS can single out certain others, which bear a definite 
relation to the first, and are known as its octave, fifth, &c. The 
corresponding differences of pitch are called intervals, and are 
spoken of as always the same for the same relationship. Thus, 
wherever they may occur in the scale, a note and its octave are 
separated by t/ie interval of the octave. It will be our object later 
to explain, so far as it can be done, the origin and nature of the 
consonant intervals, but we must now turn to consider the physical 
aspect of the question. 

Since sounds are produced by vibrations, it is natural to 
suppose that the simpler sounds, viz. musical notes, correspond to 
periodic vibrations, that is to say, vibrations which after a certain 
interval of time, called the period, repeat themselves with perfect 
regularity. And this, with a limitation presently to be noticed, 
is true. 

11. Many contrivances may be proposed to illustrate the 
generation of a musical note. One of the simplest is a revolving 
wheel whose milled edge is pressed against a card. Each 
projection as it strikes the card gives a slight tap, whose regular 
recurrence, as the wheel turns, produces a note of definite pitch, 
rising in the scale, as the velocity of rotation iivcreases. But the 
most appropriate instrument for the fundamental experiments od 
notes is undoubtedly the Siren, invented by Cagniard de la Tour. 
It consists essentially of a stiff disc, capable of revolving about its 
centre, and pierced with one or more sets of holes, arranged at 
equal intervals round the circumference of circles concentric with 
the disc. A windpipe in connection with bellows is presented 
perpendicularly to the disc, its open end being opposite to one of 
the circles, which contains a set of holes. When the bellows are 
worked, the stream of air escapes freely, if a hole is opposite to the 
end of the pipe; but otherwise it is obstructed. As the disc turns, 
a succession of puffs of air escape through it, until, when the 
velocity is sufficient, they blend into a note, whose pitch rises 
continually with the rapidity of the puffs. We shall have occasion 
later to describe more elaborate forms of the Siren, but for our 
immediate purpose the present simple arrangement will suffice. 

12. One of the most important facts in the whole science is 
exemplified by the Siren — namely, that the pitch of a note depends 
upon the period of its vibration. The size and shape of the holes, 
the force of the wind, and other elements of the problem may be 



6 INTRODUCTION. [12. 

varied ; but if the number of puffs in a given time, such as one 
second^ remains unchanged, so also does the pitch. We may even 
dispense with wind altogether, and produce a note by allowing 
the comer of a card to tap against the edges of the holes, as they 
revolve; the pitch will still be the same. Observation of other 
sources of sound, such as vibrating solids, leads to the same con- 
clusion, though the difficulties are often such as to render 
necessary rather refined experimental methods. 

But in saying that pitch depends upon period, there 
lurks an ambiguity, which deserves attentive consideration* 
as it will lead us to a point of great importance. If a 
variable quantity be periodic in any time r, it is also periodic 
in the times 2t, 3t, &c. Conversely, a recurrence within a given 
period T, does not exclude a more rapid recurrence within 
periods which are the aliqi!iot parts of t. It would appear 
accordingly that a vibration really recurring in the time ^ (for 
example) may be regarded as having the period t, and therefore 
by the law just laid down as producing a note of the pitch defined 
by T. The force of this consideration cannot be entirely evaded by 
defining as the period the least time required to bring about a 
repetition. In the first place, the necessity of such a restriction 
is in itself almost sufficient to shew that we have not got to the 
root of the matter ; for although a right to the period t may be 
denied to a vibration repeating itself rigorously within a time ^t, 
yet it must be allowed to a vibration that may differ indefinitely 
little therefi*om. In the Siren experiment, suppose that in one 
of the circles of holes containing an even number, every alternate 
hole is displaced along the arc of the circle by the same amount. 
The displacement may be made so small that no change can be 
detected in the resulting note; but the periodic time on which 
the pitch depends has been doubled. And secondly it is evident 
from the nature of periodicity, that the superposition on a vibra- 
tion of period T, of others having periods |t, Jt...&c., does not 
disturb the period t, while yet it cannot be supposed that the 
addition of the new elements has left the quality of the sound un- 
changed. Moreover, since the pitch is not affected by their 
presence, how do we know that elements of the shorter periods 
were not there from the beginning? 

13. These considerations lead us to expect remarkable rela- 
tions between the notes whose periods are as the reciprocals of the 



13.] MUSICAL INTERVALS. 7 

natural numbers. Nothing can be easier than to investigate the 
question by meaus of the Siren. Imagine two circles of holes, the 
inner containing any convenient number, and the outer twice as 
many. Then at whatever speed the disc may turn, the period of 
the vibration engendered by blowing the first set will necessarily 
be the double of that belonging to the second. Od making the 
experiment the two notes are found to stand to each other in 
the relation of octaves ; and we conclude that in passing from any 
note to its octave, the frequency, of vibration is doubled. A similar 
method of experimenting shews, that to the ratio of periods 3 : 1 
corresponds the interval known to musicians as the twelfth, made 
up of an octave and a fifth; to the ratio of 4:1, the double 
octave ; and to the ratio 5:1, the interval made up of two octaves 
and a major third. In order to obtain the intervals of the fifth 
and third themselves, the ratios must be made 3 : 2 and 5 : 4 
respectively. 

14. From these experiments it appears that if two notes 
stand to one another in a fixed relation, then, no matter at what 
part of the scale they may be situated, their periods are in a 
certain constant ratio characteristic of the relation. The same 
may be said of their freque7icies\ or the number of vibrations 
which they execute in a given time. The ratio 2 : 1 is thus 
cliaracteristic of the octave interval. If we wish to combine 
two intervals, — for instance, starting from a given note, to take 
a step of an octave and then another of a fifth in the same 
direction, the coiTesponding ratios must be compounded : 

2 3_3 
1^2""1- 

The twelfth part of an octave is represented by the ratio v 2 : 1, 
for this is the step which repeated twelve times leads to an 
octave above the starting point. If we wish to have a measure 
of intervals in the proper sense, we must take not the character- 
istic ratio itself, but the logarithm of that ratio. Then, and then 
only, will the measure of a compound interval be the suvi of the 
measures of the components. 

> A single word to denote the number of vibrations executed in the unit of time 
is indispensable : I know no better than * frequency, ' which was used in this sense 
by Toung. The same word is employed by Prof. Everett in his excellent edition 
of Deschanel's Natural Philotophy, 



8 INTRODUCTION. [15. 

16. From the intervals of the octave, fifth, and third con- 
sidered above, others known to musicians may be derived. The 
diflTerence of an octave and a fifth is called a fourth^ and has the 

3 4. 
ratio 2 ^ ^ = ^ . This process of subtracting an interval fi*om 

the octave is called inverting it. By inverting the major third 
we obtain the minor sixth. Again, by subtraction of a major 
third from a fifth we obtain the minor third ; and from this by 
inversion the major sixth. The following table exhibits side by 
side the names of the intervals and the corresponding ratios of 
frequencies : 

Octave 2 : 1 

Fifth 3:2 

Fourth 4:3 

Major Third 5 : 4 

Minor Sixth 8:5 

Minor Third 6:5 

Major Sixth 5:3 

These are all the consonant intervals comprised within the 
limits of the octave. It will be remarked that the correspouduig 
ratios are all expressed by means of small whole numbers, and 
that this is more particularly the case for the more consonant 
intervals. 

The notes whose frequencies are multiples of that of a given 
one, are called its harmonics^ and the whole series constitutes 
a harmonic scale. As is well known to violinists, they may all 
be obtained from the same string by touching it lightly with the 
finger at certain points, while the bow is drawn. 

The establishment of the connection between musical intervals 
and definite ratios of frequency — a fundamental point in Acoustics 
— is due to Mersenne (1636). It was indeed known to the 
Greeks in what ratios the lengths of strings must be changed 
in order to obtain the octave and fifth; but Mersenne demon- 
strated the law connecting the length of a string with the period 
of its vibration, and made the first determination of the actual 
rate of vibration of a known musical note. 

16. On any note taken as a key-note, or toniCy a dialonic 
scale may be founded, whose derivation we now proceed to ex- 
plain. If the key-note, whatever may be its absolute pitch, be 
called Do, the fifth above or dominant is Sol, and the fifth below 



16.] NOTATION. 9 

or snbdominant is Fa. The common chord on any note is pro- 
duced by combining it with its major third, and fifth, giving the 

ratios of frequency 1 : ^ • « or 4:5:6. Now if we take the 

common chord on the tonic, on the dominant, and on the sub- 
dominant, and transpose them when necessary into the octave 
lying immediately above the tonic, we obtain notes whose fre- 
quencies arranged in order of magnitude are : 

Do Re Mi Fa Sol La Si Do 

, 9 5 4 3 5 15 

8' 4' 3' 2' 3' 8 ' 

Here the common chord on Do is Do^Mi — Sol, with the 

5 3 

ratios 1 : t • s ; ^^^ chord on Sol is Sol — Si — Re, with the ratios 
4 2 

^:-^:2xr- = l:-:^; and the chord on Fa is Fa — La — Do, 

still with the same ratios. The scale is completed by repeating 
these notes above and below at intervals of octaves. 

If we take as our Do, or key-note, the lower c of a tenor voice, 
the diatonic scale will be 

c d e f g a b c'. 

Usage differs slightly as to the mode of distinguishing the 
different octaves ; in what follows I adopt the notation of Helm- 
holtz. The octave below the one just referred to is written with 
capital letters — C, D, &c. ; the next below that with a suffix — 
C^, D,, &c. ; and the one beyond that with a double suffix — C^^, &c. 
On the other side accents denote elevation by an octave — c', c", 
&c. The notes of the four strings of a violin are written in this 
notation, g— d'— a'— e". The middle c of the pianoforte is c'. 
[In French notation c' is denoted by ut,.] 

17. With respect to an absolute standard of pitch there has 
been no uniform practice. At the Stuttgard conference in 1834, 
c' = 264 complete vibrations per second was recommended. This 
corresponds to a' = 440. The French pitch makes a' = 435. In 
Handel's time the pitch was much lower. If c' were taken at 256 
or 2*, all the c's would have frequencies represented by powers 
of 2. This pitch is usually adopted by physicists and acoustical 
instrument makers, and has the advantage of simplicity. 

The determination ab initio of the frequency of a given note is 
an operation requiring some care. The simplest method in prin- 



1 INTRODUCTION. [ 1 7 . 

ciple is by means of the Siren, which is driven at such a rate as to 
give a note in unison with the given one. The number of turns 
effected by the disc in one second is given by a counting apparatus, 
which can be thrown in and out of gear at the beginning and end 
of a measured interval of time. This multiplied by the number of 
effective holes gives the required frequency. The consideration of 
other methods admitting of greater accuracy must be deferred. 

18. So long as we keep to the diatonic scale of c, the notes 
above written are all that are required in a musical composition. 
But it is frequently desired to change the key-note. Under these 
circumstances a singer with a good natural ear, accustomed to 
perform without accompaniment, takes an entirely fresh departure, 
coDstructing a new diatonic scale on the new key-note. In this 
way, after a few changes of key, the original scale will be quite 
departed from, and an immense variety of notes be used. On an 
instrument with fixed notes like the piano and organ such a 
multiplication is impracticable, and some compromise is necessary 
in order to allow the same note to perform different functions. 
This is not the place to discuss the question at any length; we 
will therefore take as an illustration the simplest, as well as the 
commonest case — modulation into the key of the dominant. 

By definition, the diatonic scale of c consists of the common 
chords founded on c, g and f. In like manner the scale of g con- 
sists of the chords founded on g, d and c. The chords of c and g 
are then common to the two scales ; but the third and fifth of d 
introduce new notes. The third of d written fjf has a frequency 

^ X 7 = ^ , and is far removed from any note in the scale of c. 

9 3 27 
But the fifth of d, with a frequency o >< 3 = t^ » differs but little 

from a, whose frequency is ^ . In ordinary keyed instruments the 

81 
interval between the two, represented by ^ , and called a comma, 

is neglected, and the two notes by a suitable compromise or 
temperament are identified. 

19. Various systems of temperament have been used; the 
simplest and that now most generally used, or at least aimed at, 
is the eqiml temperament. On referring to the table of frequencies 
for the diatonic scale, it will be seen that the intervals from Do to 
Re, from Re to Mi, from Fa to Sol, from Sol to La, and from La 



19.] 



EQUAL TEMPERAMENT. 



11 



9 10 
to Si, are nearly the same, being represented by - or -q- ; while the 

1 fi 
intervals from Mi to Fa and from Si to Do, represented by z-r , are 

about half as much. The equal temperament treats these ap- 
proximate relations as exact, dividing the octave into twelve equal 
parts called mean semitones. From these twelve notes the diatonic 
scale belonging to any key may be selected according to the 
following rule. Taking the key-note as the first, fill up the series 
with the third, fifth, sixth, eighth, tenth, twelfth and thirteenth 
notes, counting upwards. In this way all diflSculties of modulation 
are avoided, as the twelve notes serve as well for one key as for 
another. But this advantage is obtained at a sacrifice of true 
intonation. The equal temperament third, being the third part of 

an octave, is represented by the ratio v^2 : 1, or approximately 
1*2599, while the true third is 1*25. The tempered third is thus 
higher than the true by the interval 126 : 125. The ratio of the 
tempered fifth may be obtained from the consideration that seven 
semitones make a fifth, while twelve go to an octave. The ratio is 

7 

therefore 2^^ : 1, which = 1'4983. The tempered fifth is thus too 
low in the ratio 1-4983 : 1*5, or approximately 881 : 882. This 
error is insignificant; and even the error of the third is not of 
much consequence in quick music on instruments like the piano- 
forte. But when the notes are held, as in the harmonium and 
organ, the consonance of chords is materially impaired. 

20. The following Table, giving the twelve notes of the chro- 
matic scale according to the system of equal temperament, will be 
convenient for reference*. The standard employed is a' =440; in 





c. 


c 


16-35 


cjf 


17-32 


D 


18-36 


^ 


19-44 


E 


20-60 



c. 



c 



F 

I' 

A 



21-82 
23-12 
24-50 
25-95 
27-60 
29-13 
30-86 



32-70 
34-65 
36-71 
38-89 
41-20 
43-65 
46-25 
49-00 
61-91 
56-00 
58-27 
61-73 



"y 



jt 



jii 



65-41 
69-30 
73-42 
77-79 
82-41 
87-31 
92-50 
98-00 
103-8 
110-0 
116-5 
123-5 



130-8 
138-6 
146-8 
155-6 
164-8 
174-6 
1850 
I960 
207-6 
220-0 
2331 
246-9 



261-7 
277-2 
293-7 
311-2 
329-7 
349-2 
370-0 
392 
415-3 
440-0 
466-2 
493-9 



523-3 
544-4 
587-4 
622-3 
659-3 
698-5 
740-0 
784-0 
830-6 
880-0 
932-3 
987-7 



1046-6 
1108-8 
1174-8 
1244-6 
1318-6 
1397-0 
1480-0 
1568-0 
1661-2 
1760-0 
1864-6 
1975-5 



2093-2 
2217-7 
2349-6 
2489-3 
2637-3 
2794-0 
29601 
31360 
3322-5 
3520-0 
3729-2 
3951-0 



1 Zamminer, Die Muiik und die mu$ika1i8chen Instrumente. Giessen, 1855. 



12 



INTRODUCTION. 



[20. 



order to adapt the Table to any other absolute pitch, it is only 
necessary to multiply throughout by the proper constant. 

The ratios of the intervals of the equal temperament scale are 
given below (Zamminer) : — 



Note. 


Freqaenoy. 


Note. 


Freqaency. 


c 


= 1-00000 


f» 


21^ = 1-41421 


c« 


2^^=1-05946 


g 


2^^=1.49831 


d 


2^=1-12246 


g« 


21^-* = 1-58740 


d« 


2i-» = 118921 


a 


2^*^=1-68179 


e 


2^^=1-25992 


a« 


2ii* = 1-78180 


f 


2^=1-33484 


b 


2^^=1-88775 



c' = 2-000 



21. Returning now for a moment to the physical aspect of the 
question, we will assume, what we shall afterwards prove to be 
true within wide limits, — that, when two or more sources of sound 
agitate the air simultaneously, the resulting disturbance at any 
point in the external air, or in the ear-passage, is the simple sum 
(in the extended geometrical sense) of what would b§ caused by 
each source acting separately. Let us consider the disturbance 
due to a simultaneous sounding of a note and any or all of its 
harmonics. By definition, the complex whole forms a note having 
the same period (and therefore pitch) as its gravest element. We 
have at present no criterion by which the two can be distinguished, 
or the presence of the higher harmonics recognised. And yet— in 
the case, at any rate, where the component sounds have an inde- 
pendent origin — it is usually not difficult to detect them by the 
ear, so as to effect an analysis of the mixture. This is as much as 
to say that a strictly periodic vibration may give rise to a sensa- 
tion which is not simple, but susceptible of further analysis. In 
point of fiskct, it has long been known to musicians that under 
certain circumstances the harmonics of a note may be heard along 
with it, even when the note is due to a single source, such as a 
vibrating string ; but the significance of the fact was not under- 
stood. Since attention has been drawn to the subject, it has been 
proved (mainly by the labours of Ohm and Helmholtz) that almost 
all musical notes are highly compound, consisting in fact of the 
notes of a harmonic scale, from which in particular cases one or 



21.] NOTES AND TONES. 13 

more members may be missing. The reason of the uncertainty 
and difficulty of the analysis will be touched upon presently. 

22. That kind of note which the ear cannot further resolve is 
called by Helmholtz in German a * ton,* Tyndall and other recent 
writers on Acoustics have adopted ' tone' as an English equivalent, 
— a practice which will be followed in the present work. The 
thing is so important, that a convenient word is almost a matter 
of necessity. Notes then are in general made up of tones, the 
pitch of the note being that of the gravest tone which it contains. 

23. In strictness the quality of pitch must be attached in the 
first instance to simple tones only ; otherwise the difficulty of dis- 
continuity before referred to presents itself. The slightest change 
in the nature of a note may lower its pitch by a whole octave, as 
was exemplified in the case of the Siren. We should now rather 
say that the effect of the slight displacement of the alternate 
holes in that experiment was to introduce a new feeble tone an 
octave lower than any previously present. This is suflScient to 
alter the period of the whole, but the great mass of the sound 
remains very nearly as before. 

In most musical notes, however, the fundamental pr gravest 
tone is present in sufficient intensity to impress its character on 
the whole. The effect of the harmonic overtones is then to modify 
the quality or character^ of the note, independently of pitch. 
That such a distinction exists is well known. The notes of a violin, 
tuning fork, or of the human voice with its different vowel sounds, 
&C., may all have the same pitch and yet differ independently of 
loudness ; and though a part of this difference is due to accom- 
panying noises, which are extraneous to their nature as notes, still 
there is a part which is not thus to be accounted for. Musical 
notes may thus be classified as variable in three ways: First, pitch. 
This we have already sufficiently considered. Secondly, character, 
depending on the proportions in which the harmonic overtones are 
combined with the fundamental : and thirdly, loudness. This has 
to be taken last, because the ear is not capable of comparing 
(with any precision) the loudness of two notes which differ much 
in pitch or character. We shall indeed in a future chapter give a 
mechanical measure of the intensity of sound, including in one 
system all gradations of pitch ; but this is nothing to the point. 

1 Gennan, *Klaiigfiarbe* — French, < timbre.' The word * character' is used in 
this aenae by Everett. 



1 4 INTRODUCTION. [2 3 . 

We are here concerned with the intensity of the sensation of 
sound, not with a measure of its physical cause. The difference of 
loudness is, however, at once recognised as one of more or less ; so 
that we have hardly any choice but to regard it as dependent 
coBteria parilma on the magnitude of the vibrations concerned. 

24. We have seen that a musical note, as such, is due to a 
vibration which is necessarily periodic; but the converse, it is 
evident, cannot be true without limitation. A periodic repetition 
of a noise at intervals of a second — for instance, the ticking of a 
clock — would not result in a musical note, be the repetition ever 
so perfect In such a case we may say that the fundamental tone 
lies outside the limits of hearing, and although some of the 
harmonic overtones would fall within them, these would not give 
rise to a musical note or even to a chord, but to a noisy mass of 
sound like that produced by striking simultaneously the twelve 
notes of the chromatic scale. The experiment may be made with 
the Siren by distributing the holes quite irregularly round the 
circumference of a circle, and turning the disc with a moderate 
velocity. By the construction of the instrument, ever3rthing 
recurs after each complete revolution. 

■ 

26. The principal remaining difficulty in the theory of notes 
and tones, is to explain why notes are sometimes analysed by the 
ear into tones, and sometimes not. If a note is really complex, 
why is not the fact immediately and certainly perceived, and the 
components disentangled ? The feebleness of the harmonic over- 
tones is not the reason, for, as we shall see at a later stage of our 
inquiry, they are often of surprising loudness, and play a prominent 
part in music. On the other hand, if a note is sometimes perceived 
as a whole, why does not this happen always ? These questions 
have been carefully considered by HelmholtzS with a tolerably 
satisfactory result. The diflSculty, such as it is, is not peculiar to 
Acoustics, but may be paralleled in the cognate science of Physio- 
logical Optics. 

The knowledge of external things which we derive from the 
indications of our senses, is for the most part the result of inference. 
When an object is before us, certain nerves in our retinae are 
excited, and certain sensations are produced, which we are 
accustomed to associate with the object, and we forthwith infer its 
presence. In the case of an unknown object the process is much 

^ Tonempjindungent 3rd edition, p. 98. 



25.] ANALYSIS OF NOTES. 15 

the same. We interpret the sensations to which we are subject so 
as to form a pretty good idea of their exciting cause. From the 
slightly different perspective views received by the two eyes we 
infer, often by a highly elaborate process, the actual relief and 
distance of the object, to which we might otherwise have had no 
clue. These inferences are made with extreme rapidity and quite 
unconsciously. The whole life of each one of us is a continued 
lesson in interpreting the signs presented to us, and in drawing 
conclusions as to the actualities outside. Only so far as we succeed 
in doing this, are our sensations of any use to us in the ordinary 
affairs of life. This being so, it is no wonder that the study of our 
sensations themselves falls into the background, and that subjective 
phenomena, as they are called, become exceedingly diflScult of 
observation. As an instance of this, it is sufficient to mention the 
'blind spot' on the retina, which might a priori have been 
expected to manifest itself as a conspicuous phenomenon, though 
as a fact probably not one person in a hundred million would find 
it out for themselves. The application of these remarks to the 
question in hand is tolerably obvious. In the daily use of our ears 
our object is to disentangle from the whole mass of sound that 
may reach us, the parts coming from sources which may interest 
us at the moment. When we listen to the conversation of a friend, 
we fix our attention on the sound proceeding from him and 
endeavour to grasp that as a whole, while we ignore, as far as 
possible, any other sounds, regarding them as an interruption. 
There are usually sufficient indications to assist us in making this 
partial analysis. When a man speaks, the whole sound of his 
voice rises and falls together, and we have no difficulty in recog- 
nising its unity. It would be no advantage, but on the contrary 
a great source of confusion, if we were to carry the analysis further, 
and resolve the whole mass of sound present into its component 
tones. Although, as regards sensation, a resolution into tones 
might be expected, the necessities of our position and the practice 
of our lives lead us to stop the analysis at the point, beyond 
which it would cease to be of service in deciphering our sensa- 
tions, considered as signs of external objects^ 

But it may sometimes happen that however much we may 
wish to form a judgment, the materials for doing so are absolutely 

> Most probably the power of attending to the important and ignoring the 
unimportant part of onr sensations is to a great extent inherited — to how great an 
extent we shiJl perhaps never know. 



16 INTRODUCTION, [25. 

wanting. When a note and its octave are sounding close together 
and with perfect uniformity, there is nothing in our sensations to 
enable us to distinguish, whether the notes have a double or a 
single origin. In the mixture stop of the organ, the pressing down 
of each key admits the wind to a group of pipes, giving a note and 
its first three or four harmonics. The pipes of each group always 
sound together, and the result is usually perceived as a single 
note, although it does not proceed from a single source. 

26. The resolution of a note into its component tones is a 
matter of very different diflSculty with different individuals. A 
considerable effort of attention is required, particularly at first ; 
and, imtil a habit has been formed, some external aid in the shape 
of a suggestion of what is to be listened for, is very desirable. 

The diflSculty is altogether very similar to that of learning to 
draw. From the machinery of vision it might have been expected 
that nothing would be easier than to make, on a plane surface, a 
representation of surrounding solid objects ; but experience shews 
that much practice is generally required. 

We shall return to the question of the analysis of notes at a 
later stage, after we have treated of the vibrations of strings, with 
the aid of which it is best elucidated; but a very instructive 
experiment, due originally to Ohm and improved by Helmholtz, 
may be given here. Helmholtz^ took two bottles of the shape 
represented in the figure, one about twice as large as the other. 
These were blown by streams of air directed 
across the mouth and issuing from gutta-percha 
tubes, whose ends had been softened and pressed 
flat, so as to reduce the bore to the form of a 
narrow slit, the tubes being in connection with 
the same bellows. By pouring in water when 
the note is too low and by partially obstructing 
the mouth when the note is too high, the bottles 
may be made to give notes with the exact 
interval of an octave, such as b and b'. The 
larger bottle, blown alone, gives a somewhat muflSed sound similar 
in character to the vowel U; but, when both bottles are blown, 
the character of the resulting sound is sharper, resembling rather 
the vowel O. For a short time after the notes had been heard 
separately Helmholtz was able to distinguish them in the mixture ; 

^ Tonempfindungen^ p. 109. 




26.] PENDULOUS VIBRATIONS. 17 

but as the memory of their separate impressions &ded, the higher 
note seemed by degrees to amalgamate with the lower, which at 
the same time became louder and acquired a sharper character. 
This blending of the two notes may take place even when the high 
note is the louder. 

27. Seeing now that notes are usually compound, and that 
only a particular sort called tones are incapable of further analysis, 
we are led to inquire what is the physical characteristic of tones, 
to which they owe their peculiarity? What sort of periodic vibra- 
tion is it, which produces a simple tone? According to what 
mathematical function of the time does the pressure vary in 
the passage of the ear ? No question in Acoustics can be more 
important. 

The simplest periodic functions with which, mathematicians 
are acquainted are the circular functions, expressed by a sine or 
cosine; indeed there are no others at all approaching them in 
simplicity. They may be of any period, and admitting of no 
other variation (except magnitude), seem well adapted to produce 
simple tones. Moreover it has been proved by Fourier, that the 
most general single-valued periodic function can be resolved into 
a series of circular functions, having periods which are submultiples 
of that of the given function. Again, it is a consequence of the 
general theory of vibration that the particular type, now suggested 
as corresponding to a simple tone, is the only one capable of 
preserving its integrity among the vicissitudes which it may 
have to undergo. Any other kind is liable to a sort of physical 
analysis, one part being differently affected from another. If the 
analysis within the ear proceeded on a different principle from that 
effected according to the laws of dead matter outside the ear, 
the consequence would be that a sound originally simple might 
become compound on its way to the observer. There is no reason 
to suppose that anything of this sort actually happens. When it 
is added that according to all the ideas we can form on the subject, 
the analysis within the ear must take place by means of a physical 
machinery, subject to the same laws as prevail outside, it will be 
seen that a strong case has been made out for regarding tones as 
due to vibrations expressed by circular functions. We are not 
however left entirely to the guidance of general considerations like 
these. In the chapter on the vibration of strings, we shall see 
that in many cases theory informs us beforehand of the nature of 

B. 2 



18 INTRODUCTION. [2 



/. 



the vibration executed by a string, and in particular whether any 
specified simple vibration is a component or not Here we have 
a decisive test. It is found by experiment that, whenever according 
to theory any simple vibration is present, the corresponding tone 
can be heard, but, whenever the simple vibration is absent, then 
the tone cannot be heard. We are therefore justified in asserting 
that simple tones and vibrations of a circular type are indissolubly 
connected. This law was discovered by Ohm. 



CHAPTER 11. 



HARMONIC MOTIONS. 

28. The vibratioDs expressed by a circular function of the 
time and variously designated as simple, pendulous, or harmonic^ 
are so important in Acoustics that we cannot do better than devote 
a chapter to their consideration, before entering on the dynamical 
part of our subject. The quantity, whose variation constitutes 
the ' vibration/ may be the displacement of a particle measured 
in a given direction, the pressure at a fixed point in a fluid 
medium, and so on. In any case denoting it by u, we have 

tt = acos[ — — €j (1), 

in which a denotes the amplitude^ or extreme value of tt ; r is 
the periodic time, or period, after the lapse of which the values 
of u recur; and e determines the phase of the vibration at the 
moment from which t is measured. 

Any number of harmonic vibrations of the same period affect- 
ing a variable quantity, compound into another of the same type, 
whose elements are determined as follows : 



li =s 2a cos 



i^-') 



27rt^ . 27rt^ . 

cos — za cos € -h sm — za sin e 

T T 

27rt 
= rcos 



i^-") w. 

if r={(2acos6)^ + (2a8ine)^j* (3), 

and tan d = 2a sin 6 -rSa cose (4). 



2—2 



20 HARMONIC MOTIONS. [28. 

For example, let there be two components, 

u = acoai el + a cosi el; 

then r = {a» + a'»+2aa'co8(e-e')}* (5), 

a sin e + a' sin e' .Uv 

tang= , :j (6). 

a cos e + a cos € 

Particular cases may be noted. If the phases of the two com- 
ponents agree, 

. ,. f2irt \ 
u=^{a+a) cos ( el. 

If the phases differ by half a period, 

w = (a — a ) cos f el , 

so that if a ^a,u vanishes. In this case the vibrations are often 
said to inter/ere, but the expression is rather misleading. Two 
sounds may very properly be said to interfere, when they together 
cause silence; but the mere superposition of two vibrations 
(whether rest is the consequence, or not) cannot properly be so 
called. At least if this be interference, it is difficult to say what 
non-interference can be. It will appear in the course of this 
work that when vibrations exceed a certain intensity they no 
longer compound by mere addition; this mutual action might 
more properly be called interference, but it is a phenomenon 
of a totally different nature from that with which we are now 
dealing. 

Again, if the phases differ by a quarter or by three-quarters of 
a period, cos (e — e') = 0, and 

Harmonic vibrations of given period may be represented 
by lines drawn from a pole, the lengths of the lines being pro- 
portional to the amplitudes, and the inclinations to the phases 
of the vibrations. The resultant of any number of harmonic 
vibrations is then represented by the geometrical resultant of 
the corresponding lines. For example, if they are disposed 
symmetrically round the pole, the resultant of the lines, or 
vibrations, is zero. 

29. If we measure off along an axis of x distances pro- 
portional to the time, and take u for an ordinate, we obtain the 
harmonic curve, or curve of sines, 



29.] 



COMPOSITION. 



f2irx \ 
It = a cos I ej, 



21 



where X, called the wave-length, is written in place of r, both 
quantities denoting the range of the independent variable corre- 
sponding to a complete recurrence of the function. The harmonic 
curve is thus the locus of a point subject at once to a uniform 
motion, and to a harmonic vibration in a perpendicular direc- 
tion. In the next chapter we shall see that the vibration of a 
tuning fork is simple harmonic; so that if an excited tuning 
fork be moved with uniform velocity parallel to the line of its 
handle, a tracing point attached to the end of one of its prongs 
describes a harmonic curve, which may be obtained in a permanent 
form by allowing the tracing point to bear gently on a piece of 
smoked paper. In Fig. 2 the continuous lines are two harmonic 
curves of the same wave-length and amplitude, but of different 




phases; the dotted curve represents half their resultant, being 
the locus of points midway between those in which the two 
curves are met by any ordinate. 

30. If two harmonic vibrations of different periods coexist, 



/27rt 
iisacosf — 



- € j -f- a cos ( — 7 — el . 



The resultant cannot here be represented as a simple harmonic 
motion with other elements. If t and t' be incommensurable, the 
value of u never recurs ; but, if r and t' be in the ratio of two 
whole numbers, u recurs after the lapse of a time equal to the 
least common multiple of r and t\ but the vibration is not 
simple harmonia For example, when a note and its fifth are 
sounding together, the vibration recurs after a time equal to 
twice the period of the graver. 



22 HARMONIC MOTIONS [30. 

One case of the composition of harmonic vibrations of diflferent 
periods is worth special discussion, namely, when the diflference 
of the periods is small. If we fix our attention on the course 
of things during an interval of time including merely a few 
periods, we see that the two vibrations are nearly the same as 
if their periods were absolutely equal, in which case they would, 
as we know, be equivalent to another simple harmonic vibration 
of the same period. For a few periods then the resultant 
motion is approximately simple harmonic, but the same har- 
monic will not continue to represent it for long. The vibration 
having the shorter period continually gains on its fellow, thereby 
altering the difference of phase on which the elements of the 
resultant depend. For simplicity of statement let us suppose 
that the two components have equal amplitudes, fi^quencies 
represented by m and n, where m — n is small, and that when 
first observed their phases agree. At this moment their effects 
conspire, and the resultant has an amplitude double of that of 
the components. But after a time 1 -j- 2 (m — n) the vibration 
m will have gained half a period relatively to the other; and 
the two, being now in complete disagreement, neutralize each 
other. After a further interval of time equal to that above 
named, m will have gained altogether a whole vibration, and 
complete accordance is once more re-established. The resultant 
motion is therefore approximately simple harmonic, with an 
amplitude not constant, but varying from zero to twice that of 
the components, the frequency of these alterations being m — w. 
If two tuning forks with frequencies 500 and 501 be equally 
excited, there is every second a rise and fall of sound corre- 
sponding to the coincidence or opposition of their vibrations. 
This phenomenon is called beats. We do not here fully discuss 
the question how the ear behaves in the presence of vibrations 
having nearly equal frequencies, but it is obvious that if the motion 
in the neighbourhood of the ear almost cease for a considerable 
fraction of a second, the sound must appear to fall. For reasons 
that will afterwards appear, beats are best heard when the in- 
terfering sounds are simple tones. Consecutive notes of the 
stopped diapason of the organ shew the phenomenon very 
well, at least in the lower parts of the scale. A permanent inter- 
ference of two notes may be obtained by mounting two stopped 
organ pipes of similar construction and identical pitch side by 
side on the same wind chest. The vibrations of the two pipes 



30.] OF NEARLY EQUAL PERIOD. 23 

adjust themselves to complete opposition, so that at a little 
distance nothing can be heard^ except the hissing of the wind. 
If by a rigid wall between the two pipes one sound could be 
cut off, the other would be instantly restored. Or the balance, 
on which silence depends, may be upset by connecting the ear 
with a tube, whose other end lies close to the mouth of one of the 
pipes. 

By means of beats two notes may be tuned to unison with 
great exactness. The object is to make the beats as slow as 
possible, since the number of beats in a second is equal to the 
difference of the frequencies of the notes. Under £Etvourable 
circumstances beats so slow as one in 30 seconds may be recog- 
nised, and would indicate that the higher note gains only two 
vibrations a minute on the lower. Or it might be desired merely 
to ascertain the difference of the frequencies of two notes nearly 
in unison, in which case nothing more is necessary than to count 
the number of beats. It will be remembered that the difference 
of frequencies does not determine the interval between the two 
notes; that depends on the ratio of frequencies. Thus the 
rapidity of the beats given by two notes nearly in unison is 
doubled, when both are taken an exact octave higher. 

Analytically 

u = a cos (iirmt — e) + a' cos (2imt — e'), 

where m — n is small. 

Now cos {2imt — e) may be written 

cos [lirmt — 2^ (m — n) ^ — e'}, 
and we have 

iA = rco8(27rm^— tf) (1), 

where r^^a^-^-a'^-k- 2<w! cos {27r (m - n) ^ + e' - c} (2), 

. ^ asinc + a'sin {2'7r(m — n)^ + €'} .^. 

tan O = ; ; r^ — ; 77— — n W/« 

a cos € + a cos [2'7r (m — n) ^ + e } 

The resultant vibration may thus be considered as harmonic 
with elements r and 0, which are not constant but slowly varying 
functions of the time, having the frequency m—n. The ampli- 
tude r is at its maximum when 

cos {27r (m — n) ^ + e' — €} = + 1, 
and at its minimum when 

cos {2'7r (m — n) t +€' — €}=— 1, 
the corresponding values being a-^-a' and a^a respectively. 



24 HAKMONIC JfOTIONS. [31. 

31. Another case of great importance is the composition of 
vibrations corresponding to a tone and its harmonica It is known 
that the most general single-valued finite periodic function can 
be expressed by a series of simple harmonics — 

/2imt 



U = ao + 2^l On cos f-^j; €«) (1), 



a theorem usually quoted as Fourier's. Analytical proofs will be 
found in Todhunter's Integral Calcvlus and Thomson and Tait's 
Natural Philosophy ; and a line of argument almost if not quite 
amounting to a demonstration will be given later in this work. 
A few remarks are all that will be required here. 

Fourier's theorem is not obvious. A vague notion is not un- 
common that the infinitude of arbitrary constants in the series 
of necessity endows it with the capacity of representing an arbi- 
trary periodic function. That this is an error will be apparent, 
when it is observed that the same argument would apply equally, 
if one term of the series were omitted ; in which case the ex- 
pansion would not in general be possible. 

Another point worth notice is that simple harmonics are not 
the only functions, in a series of which it is possible to expand 
one arbitrarily given. Instead of the simple elementary term 

"""^Vl- — ^V' 

we might take 

cos I €« 1 + X cos ( €n 






formed by adding a similar one in the same phase of half the 
amplitude and period. It is evident that these terms would 
serve as well as the others; for 



iTTvt \ f /2irnt \ 1 



2iT7vt \ f f^irrd \ 1 fAsimt 

cos I 6«| = -IcOSf €nl4-T^C0S f €n 



1 ( (4nmt \ 1 (Simt 
" 2 P^ V^ «nj + 2 ^^^ \;^ ^ 

— ad infin., 

so that each term in Fourier's series, and therefore the sum of 
the series, can be expressed by means of the double elementary 



31.] Fourier's theorem. 25 

terms now suggested. This is mentioned here, because students, 
not being acquainted with other expansions, may imagine that 
simple harmonic functions are by nature the only ones qualified 
to be the elements in the development of a periodic function. 
The reason of the preeminent importance of Fourier's series in 
Acoustics is the mechanical one referred to in the preceding 
chapter, and to be explained more fully hei-eafter, namely, that, 
in general, simple harmonic vibrations are the only kind that are 
propagated through a vibrating system without suffering decom- 
position. 

32. As in other cases of a similar character, e.g. Taylor's 
theorem, if the possibility of the expansion be known, the co- 
efficients may be determined by a comparatively simple process. 
We may write (1) of § 81 



w = ilo + 2^i -4»cos 1-2^1 -DnSin — (1). 

T T 

Multiplying by cos {^mrtjr) or sin (2n7rt/T), and integrating 
over a complete period from ^ = to ^ = t, we find 



. 2 r^ 2n7rt ,/ 

-^n^- \ UCO& at 

T J 



T 



Bn=- I usin — — at 

r J r 



(2). 



An immediate integration gives 

-4o = - rudt (3), 

T J 

indicating that Aq ia the mean value of v throughout the period. 

The degree of convergency in the expansion of u depends in 
general on the continuity of the function and its derivatives. 
The series formed by successive differentiations of (1) converge 
less and less rapidly, but still remain convergent, and arithmetical 
representatives of the differential coefficients of u, so long as 
these latter are everywhere finite. Thus (Thomson and Tait, 
§ 77), if all the derivatives up to the m^ inclusive be free 
finom infinite values, the series for u is more convergent than 
one with 



for coefficients. 



1 ^ ^ ^ &c 
2^' S^' 4"*' *' 



26 HARMONIC MOTIONS [32 a. 

32 a. The general explanation of the beats heard when two 
pure tones nearly in unison are sounded simultaneously has been 
discussed in § 30. But the occurrence of beats is not confined to 
the case of approximate unison, at least when we have to deal 
with compound notes. Suppose for example that the interval 
is an octava The graver note then usually includes a tone 
coincident in pitch with the fundamental tone of the higher note. 
If the interval be disturbed, the previously coincident tones 
sepstrate from one another, and give rise to beats of the same 
frequency as if they existed alone. There is usually no difficulty 
in observing these beats; but if one or both of the component 
tones concerned be very faint, the aid of a resonator may be 
invoked. 

In general we may consider that each consonant interval is 
characterized by the coincidence of certain component tones, and 
if the interval be disturbed the previously coincident tones 
give rise to beats. Of course it may happen in any particular 
case that the tones which would coincide in pitch are absent from 
one or other of the notes. The disturbance of the interval 
would then, according to the above theory, not be attended 
by beats. In practice faint beats are usually heard; but the 
discussion of this phenomenon, as to which authorities are not 
entirely 'agreed, must be postponed. 

33. Another class of compounded vibrations, interesting from 
the fecility with which they lend themselves to optical observa- 
tion, occur when two harmonic vibrations affecting the same par- 
ticle are executed in perpendicular directions, more especially 
when the periods are not only commeasurable, but in the ratio 
of two small whole numbers. The motion is then completely 
periodic, with a period not many times greater than those of the 
components, and the curve described is re-entrant. If u and v 
be the co-ordinates, we may take 

w = a cos (27m^ — €), v = 6cos27m'^ (1). 

First let us suppose that the periods are equal, so that n' ^n\ 
the elimination of t gives for the equation of the curve described, 

^+p-"^'°«^-^^=^ <2>' 

representing in general an ellipse, whose position and dimensions 
depend upon the amplitudes of the original vibrations and upon 



33.] IN PERPENDICULAR DIRECTIONS. 27 

the difference of their pbasea If the phases differ by a quarter 
period, cos € « 0, and the equation becomes 

"i + ji = 1- 

In this case the axes of the ellipse coincide with those of 
co-ordinates. If further the two components have equal ampli- 
tudes, the locus degenerates into the circle 

w* + v* = a', 

which is described with uniform velocity. This shews how a 
uniform circular motion may be analysed into two rectilinear 
harmonic motions, whose directions are perpendicular. 

If the phases of the components agree, € — 0, and the ellipse 
degenerates into the coincident straight lines 

or if the difference of phase amount to half a period, into 



(M)"-"- 



When the unison of the two vibrations is exact, the elliptic 
path remains perfectly steady, but in practice it will almost 
always happen that there is a slight difference between the 
periods. The consequence ia that though a fixed ellipse represents 
the curve described with sufficient accuracy for a few periods, 
the ellipse itself gradually changes in correspondence with the 
alteration in the magnitude of e. It becomes therefore a matter 
of interest to consider the system of ellipses represented by (2), 
supposing a and b constants, but e variable. 

Since the extreme values of u and v are ± a, ± 6 respectively, 
the ellipse is in all cases inscribed in the rectangle whose sides 
are 2a, 26. Starting with the phases in agreement, or 6 = 0, we 

have the ellipse coincident with the diagonal — a"^* ^ 

€ increases from to ^tt, the ellipse opens out until its equation 
becomes 



y} v* 



i + j5 = 1- 
From this point it closes up again, ultimately coinciding with 

the other diagonal - + ^ = 0, corresponding to the increase of e from 

^ to TT. After this, as e ranges ftt)m v to 27r, the ellipse retraces 



28 



HARMONIC MOTIONS. 



[33. 



its course until it again coincides with the first diagonal. The 
sequence of changes is exhibited in Fig. 3. 

F/G.3, 





' — 







The ellipse, having already four given tangents, is completely 
determined by its point of contact P (Fig. 4) with the line v = 6. 



F/ G. -4. 




In order to connect this with e, it is sufficient to observe that 
when V = 6, cos ^irvi = 1 ; and therefore u = a cos 6. Now if the 
elliptic paths be the result of the superposition of two harmonic 
vibrations of nearly coincident pitch, € varies uniformly with the 
time, so that P itself executes a harmonic vibration along A A' 
with a frequency equal to the difference of the two given fre- 
quenciea 

34L Idssajous^ has shewn that this system of ellipses may be 
regarded as the different aspects of one and the same ellipse 
described on the surface of a transparent cylinder. In Fig. 5 



A 




A 








b' 




B 


^ Anm 


r/G. 5 
iU$ de Chitnie (8) li. 147, 


1S67. 



34.] 



USSAJOUS CYLINDER. 



29 



AA'BB represents the cylinder, of which AB! is a plane section. 
Seen from an infinite distance in the direction of the conunon 
tangent at il to the plane sections, the cylinder is projected into a 
rectangle, and the ellipse into its diagonal. Suppose now that the 
cylinder turns upon its axis, carrying the plane section with it. 
Its own projection remains a constant rectangle in which the pro- 




F/G. 6. 

jection of the ellipse is inscribed. Fig. 6 represents the posi- 
tion of the cylinder after a rotation through a right angle. It 
appears therefore that by turning the cylinder round we obtain in 
succession all the ellipses corresponding to the paths described by 
a point subject to two harmonic vibrations of equal period and fixed 
amplitudes. Moreover if the cylinder be turned continuously 
with uniform velocity, which insures a harmonic motion for P, 
we obtain a complete representation of the varpng orbit de- 
scribed by the point when the periods of the two components 
differ slightly, each complete revolution answering to a gain or 
loss of a single vibration^ The revolutions of the cylinder are 
thus synchronous with the beats which would result from the 
composition of the two vibrations, if they were to act in the same 
direction. 

36. Vibrations of the kind here considered are very easily 
realized experimentally. A heavy pendulum-bob, hung frx)m a 
fixed point by a long wire or string, describes ellipses under the 
action of gravity, which may in particular cases, according to the 
circumstances of projection, pass into straight lines or circles. 
But in order to see the orbits to the best advantage, it is necessary 
that they should be described so quickly that the impression 
on the retina made by the moving point at any part of its course 
has not time to fade materially, before the point comes round again 
to renew its action. This condition is fulfilled by the vibration of 
a silvered bead (giving by reflection a luminous point), which is 

^ By a vibration wiU always be meant in this work a complete oyole of changes. 



30 



HARMONIC MOTIONS. 



[35. 



attached to a straight metallic wire (such as a knitting-needle)* 
firmly clamped in a vice at the lower end. When the system is set 
into vibration, the luminous point describes ellipses, which appear 
as fine lines of light. These ellipses would gradually contract in 
dimensions under the influence of friction until they subsided 
into a stationary bright point, without undergoing any other 
change, were it not that in all probability, owing to some want 
of symmetry, the wire has slightly differing periods according to 
the plane in which the vibration is executed. Under these cir- 
cumstances the orbit is seen to undergo the cycle of changes 
already explained. 

36. So far we have supposed the periods of the component 
vibrations to be equal, or nearly equal ; the next case in order of 
simplicity is when one is the double of the other. We have 

u = a cos (4nnrt — e), v = 6 cos 2w7rf . 

The locus resulting firom the elimination of t may be written 



u 

- = cos e 

a 



(2^.-l) + 2sin.yi-^ (1). 



which for all values of e represents a curve inscribed in the rect- 
angle 2a, 26. If 6 = 0, or tt, we have 



-K'^S. 



r/ff.z 













36.] 



CONSONANT INTERVALS, 



31 



representing pstrabolaa Fig. 7 shews the various curves for the 
intervals of the octave, twelfth, and fifth. 

To all these systems Lissajous' method of representation by 
the transparent cylinder is applicable, and when the relative 
phase is altered, whether from the different circumstances of 
projection in different cases, or continuously owing to a slight 
deviation from exactness in the ratio of the periods, the cylinder 
will appear to turn, so as to present to the eye different aspects of 
the same line traced on its surface. 



37. There is no difficulty in arranging a vibrating system so 
that the motion of a point shall consist of two harmonic vibrations 
in perpendicular planes, with their periods in any assigned ratio. 
The simplest is that known as Blackburn's pendulum. A wire 
ACB is fastened at A and B, two fixed points at the same level. 
The bob P is attached to its middle point by another wire CP. 
For vibrations in the plane of the diagram, the point of suspension 
is practically (7, provided that the wires are sufficiently stretched ; 
but for a motion perpendicular to this plane, the bob turns about 
D, carrying the wire ACB with it. The periods of vibration in 




Op 

the principal planes are in the ratio of the square roots of CP and 
DP. Thus if DC = 3CP, the bob describes the figures of the 
octava To obtain the sequence of curves corresponding to 
approximate unison, ACB must be so nearly tight, that CD is 
relatively small 



32 HARMONIC MOTIONS. [38. 

38. Another contrivance called the kaleidophone was origin- 
ally invented by Wheatstone. A straight thin bw of steel carrying 
a bead at its upper end is fastened in a vice, as explained in a 
previous paragraph. If the section of the bar is square, or circular, 
the period of vibration is independent, of the plane in which it is 
performed. But let us suppose that the section is a rectangle 
with unequal sides. The stifihess of the bar — the force with 
which it resists bending — is then greater in the plane of greater 
thickness, and the vibrations in this plane have the shorter period. 
By a suitable adjustment of the thicknesses, the two periods of 
vibration may be brought into any required ratio, and the cor- 
responding curve exhibited. 

The defect in this arrangement is that the same bar will give 
only one set of figures. In order to overcome this objection 
the following modification has been devised. A slip of steel is 
taken whose rectangular section is very elongated, so that as 
regards bending in one plane the stiffness is so great as to amount 
practically to rigidity. The bar is divided into two parts, and the 
broken ends reunited, the two pieces being turned on one another 
through a right angle, so that the plane, which contains the small 
thickness of one, contains the great thickness of the other. When 
the compound rod is clamped in a vice at a point below the junc- 
tion, the period of the vibration in one direction, depending almost 
entirely on the length of the upper piece, is nearly constant ; but 
that in the second direction may be controlled by varjring the 
point at which the lower piece is clamped. 

39. In this arrangement the luminous point itself executes 
the vibrations which are to be observed ; but in Lissajous' form of 
the experiment, the point of light remains really fixed, while its 
linage is thrown into apparent motion by means of successive 
reflection from two vibrating mirrors. A small hole in an opaque 
screen placed close to the flame of a lamp gives a point of light,, 
which is observed after reflection in the mirrors by means of a 
small telescope. The mirrors, usually of polished steel, are attached 
to the prongs of stout tuning forks, and the whole is so disposed 
that when the forks are thrown into vibration the luminous point 
appears to describe harmonic motions in perpendicular directions, 
owing to the angular motions of the reflecting surfaces. The 
ampUtudes and periods of these harmonic motions depend upon 
those of the corresponding forks, and may be made such as to give 



39.] OPTICAL METHODS. 33 

with enhanced brilliancy any of the figures possible with the 
kaleidophone. By a similar arrangement it is possible to project 
the figures on a screen. In either case they gradually contract as 
the vibrations of the forks die away. 

40. The principles of this chapter have received an important 
application in the investigation of rectilinear periodic motions. 
When a point, for instance a particle of a sounding string, is 
vibrating with such a period as to give a note within the limits of 
hearing, its motion is much too rapid to be followed by the eye ; 
so that, if it be required to know the character of the vibration, 
some indirect method must be adopted. The simplest, theo- 
retically, is to compound the vibration under examination with a 
uniform motion of translation in a perpendicular direction, as when 
a tuning-fork draws a harmonic curve on smoked paper. Instead 
of moving the vibrating body itself, we may make use of a revolv- 
ing mirror, which provides us with an iiiuige in motion. In this 
way we obtain a representation of the function characteristic of 
the vibration, with the abscissa proportional to time. 

But it often happens that the application of this method would 
be difficult or inconvenient. In such cases we may substitute for 
the uniform motion a harmonic vibration of suitable period in the 
S€une direction. To fix our ideas, let us suppose that the point, 
whose motion we wish to investigate, vibrates vertically with a 
period T, and let us examine the result of combining with this a 
horizontal harmonic motion, whose period is some multiple of t, 
say, m. Take a rectangular piece of paper, and with axes parallel 
to its edges draw the curve representing the vertical motion (by 
setting off abscissae proportional to the time) on such a scale that 
the paper just contains n repetitions or waves, and then bend the 
paper round so as to form a cylinder, with a re-entrant curve run- 
ning round it. A point describing this curve in such a manner 
that it revolves uniformly about the axis of the cylinder will 
appear from a distance to combine the given vertical motion of 
period T, with a horizontal harmonic motion of period /it. Con- 
versely therefore, in order to obtain the representative curve of 
the vertical vibrations, the cylinder containing the apparent path 
must be imagined to be divided along a generating line, and 
developed into a plane. There is less difficulty in conceiving the 
cylinder and the situation of the curve upon it, when the adjust- 
ment of the periods is not quite exact, for then the cylinder 



34 HARMONIC MOTIONS. [40. 

appears to turn, and the contrary motions serve to distinguish 
those parts of the curve which lie on its nearer and further face. 

41. The auxiliary harmonic motion is generally obtained 
optically, by means of an instrument called a vibration-microscope 
invented by Lissajous. One prong of a large tuning-fork carries 
a lens, whose axis is perpendicular to the direction of vibration; 
and which may be used either by itself, or as the object-glass of 
a compound microscope formed by the addition of an eye-piece 
independently suppoi*ted. In either case a stationary point is 
thrown into apparent harmonic motion along a line parallel to 
that of the fork's vibration. 

The vibration-microscope may be applied to test the rigour 
and universality of the law connecting pitch and period. Thus 
it will be found that any point of a vibrating body which gives 
a pure musical note will appear to describe a re-entrant curve, 
when examined with a vibration-microscope whose note is in 
strict unison with its own. By the same means the ratios of 
frequencies characteristic of the co^isonant intervals may be 
verified; though for this latter purpose a more thoroughly 
acoustical method, to be described in a future chapter, may be 
preferred. 

42. Another method of examining the motion of a vibrating 
body depends upon the use of intermittent illumination ^ Suppose, 
for example, that by means of suitable apparatus a series of 
electric sparks are obtained at regular intervals t. A vibrating 
body, whose period is also t, examined by the light of the sparks 
must appear at rest, because it can be seen only in one position. 
If, however, the period of the vibration differ from t ever sa 
little, the illuminated position varies, and the body will appear 
to vibrate slowly with a frequency which is the difference of that 
of the spark and that of the body. The type of vibration can 
then be observed with facility. 

The series of sparks can be obtained from an induction-coil,, 
whose primary circuit is periodically broken by a vibrating fork,, 
or by some other interrupter of sufficient regularity. But a better 
result is afforded by sunlight rendered intermittent with the aid of 
a fork, whose prongs carry two small plates of metal, parallel to 
the plane of vibration and close together. In each plate is a slit' 

1 Plateau, Bull, de VAcad. roy, de Belgique, t. iii, p. 364, 1836. 



42.] INTERMITTENT ILLITMINATION. 35 

parallel to the prongs of the fork, and so placed as to afford a 
free passage through the plates when the fork is at rest, or passing 
through the middle point of its vibrations. On the opening so 
formed, a beam of sunlight is concentrated by means of a burning- 
glass, and the object under examination is placed in the cone of 
rays diverging on the further side'. When the fork is made to 
vibrate by an electro-magnetic arrangement, the illumination is cut 
off except when the fork is passing through its position of equi- 
librium, or nearly so. The flashes of light obtained by this method 
are not so instantaneous as electric sparks (especially when a 
jar is connected with the secondary wire of the coil), but in my 
experience the regularity is more perfect. Care should be taken 
to cut off extraneous light as far as possible, and the effect is then 
very striking. 

A similar result may be arrived at by lor>king at the vibrating 
body through a series of holes arranged in a circle on a revolving 
disc. Several series of holes may be provided on the same 
disc, but the observation is not satisfactory without some pro- 
vision for securing uniform rotation. 

Except with respect to the sharpness of definition, the result is 
the same when the period of the light is any multiple of that of 
the vibrating body. This point must be attended to when the 
revolving wheel is used to determine an unknown frequency. 

When the frequency of intermittence is an exact multiple of 
that of the vibration, the object is seen without apparent motion, 
but generally in more than one position. This condition of things 
is sometimes advantageous. 

Similar effects arise when the frequencies of the vibrations 
and of the flashes are in the ratio of two small whole numbers. 
If, for example, the number of vibrations in a given time be half 
as great again as the number of flashes, the body will appear 
stationary, and in general double. 

42 a. We have seen (§ 28) that the resultant of two isoperiodic 
vibrations of equal amplitude is wholly dependent upon their phase 
relation, and it is of interest to inquire what we are to expect 
from the composition of a large number (n) of equal vibrations 
of amplitude unity, of the same period, and of phases accidentally 
determined. The intensity of the resultant, represented by the 
square of the amplitude § 245, will of course depend upon the 

1 Topler, Phii. Mag. Jan. 1867. 

3—2 



36 HARMONIC MOTIONS. [42 a. 

precise manner in which the phases are distributed, and may vary 
from n* to zero. But is there a definite intensity which becomes 
more and more probable when n is increased without limit ? 

The nature of the question here i-aised is well illustrated by 
the special case in which the possible phases are restricted to two 
opposite phases. We may then conveniently discard the idea of 
phase, and regard the amplitudes as at random positive or negative. 
If all the signs be the same, the intensity is n* ; if, on the other 
hand, there be as many positive as negative, the result is zero. 
But although the intensity may range from to n^ the smaller 
values are more probable than the greater. 

The simplest part of the problem relates to what is called in 
the theory of probabilities the '* expectation " of intensity, that 
is, the mean intensity to be expected after a great number of 
trials, in each of which the phases are taken at random. The 
chance that all the vibrations are positive is (J)**, and thus the 
expectation of intensity corresponding to this contingency is 
(J)**.n'. In like manner the expectation corresponding to the 
number of positive vibrations being (n— 1) is 

and so on. The whole expectation of intensity is thus 



^l.n^ + nin-2y + '^^kn-*y 



+ "^"7^,>;^-'> (n-6)' + ...} (1). 



Now the sum of the (n + 1) terms of this series is simply n, as 

may be proved by comparison of coefficients of a;" in the equivalent 

forms 

(e* + e-«)~ = 2~ (1 + ia;» + . . .)« 

The expectation of intensity is therefore n, and this whether n be 
great or small. 

The same conclusion holds good when the phases are unre- 
stricted. From (3) § 28, if Oi = Oj = . . . = 1, 

r' = (cosei +COS €a+ ...)'-l-(sin ei + sin€,-h ..,y 

= nH- 2S cos (€, - 6i) (2), 

where under the sign of summation are to be included the cosines 
of the iw(n — 1) differences of phase. When the phases are 



42 a.] PHASES AT RANDOM. 37 

accidental, the sum is as likely to be positive as negative, and 
thus the mean value of r^ is n. 

The reader must be on his guard here against a fallacy which 
has misled some eminent authors^ We have not proved that when 
n is large there is any tendency for a single combination to give 
an intensity equal to n, but the quite different proposition that in 
a large number of trials, in each of which the phases are dis- 
tributed at random, the viean intensity will tend more and more 
to the value n. It is true that even in a single combination there 
is no reason why any of the cosines in (2) should be positive 
rather than negative. From this we may infer that when n is 
increased the sum of the terms tends to vanish in comparison with 
the number of terms ; but, the number of the terms being of the 
order n*, we can infer nothing as to the value of the sum of the 
series in comparison with n. 

So fietr there is no difficulty; but a complete investigation of 
this subject involves an estimate of the relative probabilities of 
resultants lying within assigned limits of magnitude. For example, 
we ought to be able to say what is the probability that the 
intensity due to a large number (ti) of equal components is less 
than \n. This problem may conveniently be considered here, though 
it is naturally beyond the reach of elementary methods. We will 
commence by taking it under the restriction that the phases are 
of two opposite kinds only.. 

Adopting the statistical method of statement, let us suppose 
that there are an immense number N of independent combinations, 
each consisting of n unit vibrations, positive or negative, and com- 
bined accidentally. When N is sufficiently large, the statistics 
become regular; and the number of combinations in which the 
resultant amplitude is found equal to x may be denoted by 
N ,f(n, x\ where /is a definite function of n and x. Now suppose 
that each of the N combinations receives another random contri- 
bution of ± 1, and inquire how many of them will subsequently 
possess a resultant x. It is clear that those only can do so which 
originally had amplitudes a? — 1, or a;+l. if aZ/ of the former, 
and half of the latter number will acquire the amplitude x, so 
that the number required is 

ii\r/(n, a; -l) + Ji\r/(r2,^ -hi). 
But this must be identical with the number corresponding to 
n + 1 and a?, so that 

/(n-h 1, x)^\f{n,x^\) + i/(n, ^+1) (3). 



38 HARMONIC MOTIONS. [42 a. 

This equation of differences holds good for all integral values 
of X and for all positive integral values of n. If / (n, a?) be given 
for one value of n, the equation suflSces to determine / (n, x) for 
all higher integral values of n. For the present purpose the 
initial value of n is zero. In that case we know that /(a?) = for 
all values of x other than zero, and that when x = 0,/(0, 0) = 1. 

The problem proposed in the above form is perfectly definite ; 
but for our immediate object it suflSces to limit ourselves to the 
supposition that n is great, regarding /(n, ^) as a continuous 
function of continuous variables n and x^ much as in the analogous 
problem of §§120, 121.122. 

Writing (3) in the form 

/(n + 1, x)^f(n, x) = i/(n, X- 1)+ i/*(n, x+ !)-/(% x)...(4), 

we see that the left-hand member may then be identified with 
dfldn, and the right-hand member with ^d^fjda^, so that under 
these circumstances the differential equation to which (3) reduces 
is of the well-known form 

dn" Ida? ^ ^' 

The analogy with the conduction of heat is indeed very close ; 
and the methods developed by Fourier for the solution of problems 
in the latter subject are at once applicable. The special condition 
here is that initially, that is when n = 0, / must vanish for all 
values of x other than zero. As may be verified by differentiation, 
the special solution of (5) is then 

f(n, ^) = Ae-^/.n , (6), 

in which A is an arbitrary constant to be determined from the 
consideration that the whole number of combinations is N, Thus, 
if dx be large in comparison with unity, the number of combina- 
tions which have amplitudes between x and x-\-dx\& 

--T-er^f^dx: 

while ^ e^i^dx=-N, 

so that in virtue of the known equality 

e"'^ dz = V'T, 



r+oo 

J —00 



A . V2^ = 1. 



42 a.] PHASES AT RANDOM. 39 

The final result for the number of combinations which have 
Amplitudes between x and x-\-dx\& accordingly 

V(27ni) 
The mean intensity is expressed by 

1 r"*"* 

as before. 

We will now pass on to the more important problem in which 
the phases of the n unit vibrations are distributed at random over 
the entire period. In each combination the resultant amplitude 
is denoted by r and the phase (referred to a given epoch) by ; 
and rectangular coordinates are taken so that 

a? = r cos 0, y^r sin 0. 
Thus any point (a?, y) in the plane of reference represents a 
vibration of amplitude r and phase 0, and the whole system of 
N vibrations is represented by a distribution of points, whose 
density it is our object to determine. Since no particular phase 
can be singled out for distinction, we know beforehand that the 
density of distribution will be independent of 0. 

Of the infinite number N of points we suppose that 

Nf{n, X, y) dxdy 

are to be found within the infinitesimal area dxdy, and we will 

inquire as before how this number would be changed by the 

addition to the n component vibrations of one more unit vibration 

of accidental phase. Any vibration which after the addition is 

represented by the point x, y must before have corresponded to 

the point 

of = x — cos (f>, y' ^y — sin 0, 

where represents the phase of the additional unit vibration. 
And, if for the moment be regarded as given, to the area dxdy 
corresponds an equal area dx'dy\ Again, all values of <f> being 
equally probable, the factor necessary under this head is d<l>/2'n: 
Accordingly the whole number to be found in dxdy after the 
superposition of the additional unit is 

Ndxdy rf(n, x\ y') d<f>/27r ; 

and this is to be equated to 

Ndxdy f(n + 1, x, y) ; 

so that /(w+1, X, y)^^ f{n, x\ y')d<f>l2ir (8). 



40 HARMONIC MOTIONS. [42 a. 

The value of /(w, x, y) is obtained by introduction of the 
values of x\ y and expansion: 

/(-'.3/')=/(-.y)-fcos^-|8in^ + |gco8«^ 

+ j J- cos^sin^+5 T^,8m»^+..., 
dxdy 2dy^ 

so that 

Also, n being very great, 

/(n + 1, X, y) -/(w, ar, y) = d//dn ; 
and (8) reduces to 

dn"4 W dj^y ^ ^' 

the usual equation for the conduction of heat in two dimensions. 

In addition to (9), /has to satisfy the special condition of 

evanescence when n = for all points other than the origin. The 

appropriate solution is necessarily symmetrical round the origin, 

and takes the form 

/(n, a;, y) = ^n-ie-'*'+y*»/'* (lO), 

as may be verified by differentiation. The constant -4 is to be 
determined by the condition that the whole number is N, Thus 

iV = NAvr"^ 11 e-<*'+y*»/» dxdy^ NA 2im-^r e'^l'^rdr = irAN ; 

and the number of vibrations within the area dxdy becomes 

— e-^^l^'dxdy (11). 

If we wish to find the number of vibrations which have 

amplitudes between r and r + dr, we must introduce polar 

coordinates and integrate with i-espect to 0. The required number 

is thus 

2Nn'-'e-^l''rdr (12)^ 

The result may also be expressed by saying that the probahility 
of a resultant amplitude between r and r + dr when a large 
number n of unit vibrations are compounded at random is 

2n-'e-^l''rdr (13). 

^ Phil. Mag. Aag. 1880. 



42 a.] 



PHASES AT RANDOM. 



41 



The mean intensity is given by 










as was to be expected. 

The probability of a resultant amplitude less than r is 



Jo 



2n-» e-'^^'^rdr = 1 - e-^f" 



(14), 



or, which is the same thing, the probability of a resultant ampli- 
tude greater than r is 

c-^'/~ (15). 

The following table gives the probabilities of intensities less 
than the fractions of n named in the first column. For example, 
the probability of intensity less than n is *6321. 



•05 


•0488 


1 -80 


•5506 


•10 


•0952 


100 


•6321 


•20 


•1813 


i 1-50 


•7768 


•40 


•3296 


2 00 


•8647 


•60 


•4512 


1 3^00 


•9502 



It will be seen that, however great n may be, there is a 
reasonable chance of considerable relative fluctuations of intensity 
in different combinations. 

If the amplitude of each component be a, instead of unity, as 
we have hitherto supposed for brevity, the probability of a resultant 
amplitude between r and r + dr is 



no? 



(16). 



The result is thus a function of n and a only through na^ and 
would be unchanged if for example the amplitude became \ol and 
the number 4n. From this it follows that the law is not altered, 
even if the components have different amplitudes, provided always 
that the whole number of each kind is very great; so that if there 
be n components of amplitude a, n' of amplitude ^, and so on, the 
probability of a resultant between r and r + dr is 

nar + w /CP -f ... 

That this is the case may perhaps be made more clear by the 
consideration of a particular case. Let us suppose in the first 
place that n-\-in' unit vibrations are compounded at random. 



42 * HARMONIC MOTIONS. [42 a. 

The appropriate law is given at once by (13) on substitution of 
n + 4w' for n, that is 

2 (n + 4n')-' c-^'/<«"^«Vdr (18). 

Now the combination of n + 4n' unit vibrations may be re- 
garded as arrived at by combining a random combination of n 
unit vibrations with a second random combination of 4n' units, 
and the second random combination is the same as if due to a 
random combination of n' vibrations each of amplitude 2. Thus 
(18) applies equally well to a random combination of (n + n') 
vibrations, n of which are of amplitude unity and n' of ampli- 
tude 2. 

Although the result has no application to the theory of vibra- 
tions, it may be worth notice that a similar method applies to the 
composition in three dimensions of unit vectors, whose directions 
are accidental. The equation analogous to (8) gives in place of 

(9) 

dn eKdaf' dy^'^ dzV' 
The appropriate solution, analogous to (13), is 



V UnV 



e-r«/!«r«dr (18), 



expressing the probability of a resultant amplitude lying between 
r and r + dr. 

Here again the mean value of r", to be expected in a great 
number of independent combinations, is n. 



CHAPTER III. 

SYSTEMS HAVING ONE DEGREE OF FREEDOM. 

43. The material systems, with whose vibrations Acoustics is 
concerned, are usually of considerable complication, and are sus- 
ceptible of very various modes of vibration, any or all of which 
may coexist at any particular moment. Indeed in some of the 
most important musical instruments, as strings and organ-pipes, 
the number of independent modes is theoretically infinite, and 
the consideration of several of them is essential to the most prac- 
tical questions relating to the nature of the consonant chords. 
Cases, however, often present themselves, in which one mode is 
of paramount importance ; and even if this were not so, it would 
still be proper to commence the consideration of the general 
problem with the simplest case — that of one degree of freedom. 
Jt need not be supposed that the mode treated of is the only one 
possible, because so long as vibrations of other modes do not occur 
their possibility under other circumstances is of no moment. 

41. The condition of a system possessing one degree of free- 
dom is defined by the value of a single co-ordinate u, whose origin 
may be taken to correspond to the position of equilibrium. The 
kinetic and potential energies of the system for any given position 
are proportional respectively to ti* and u' : — 

T = ^mu\ V=-^fiu* (1). 

where m and fi are in general functions of u. But if we limit 
ourselves to the consideration of positions in the immediate neigh" 
bowrhood of tfiat corresponding to equilibrium, u is a small quantity, 
and m and fi are sensibly constant. On this underatanding we 



44 ONE DEGREE OF FREEDOM. [44. 

now proceed. If there be no forces, either resulting from internal 
friction or viscosity, or impressed on the system from without, the 
whole energy remains constant. Thus 

T-f. F= constant. 

Substituting for T and V their values, and differentiating with 
respect to the time, we obtain the equation of motion 

mit-f ^w = (2) 

of which the complete integral is 

u^a cos (nt — a) (3), 

where n=*=ft-7-m, representing a harmonic vibration. It will be 
seen that the period alone is determined by the nature of the 
system itself; the amplitude and phase depend on collateral cir- 
cumstances. If the differential equation were exact, that is to 
say. if T were strictly proportional to li", and V to w', then, without 
any restriction, the vibrations of the system about its configuration 
of equilibrium would be accurately harmonic. But in the majority 
of cases the proportionality is only approximate, depending on an 
assumption that the displacement u is always small — how small 
depends on the nature of the particular system and the degree of 
approximation required ; and then of course we must be careful 
not to push the application of the integral beyond its proper 
limits. 

But, although not to be stated without a limitation, the prin- 
ciple that the vibrations of a system about a configuration of 
equilibrium have a period depending on the structure of the 
system and not on the particular circumstances of the vibration, 
is of supreme importance, whether regarded from the theoretical 
or the practical side. If the pitch and the loudness of the note 
given by a musical instrument were not within wide limits in- 
dependent, the art of the performer on many instruments, such 
as the violin and pianoforte, would be revolutionized. 

The periodic time 

. = 2^ = 2^^* (4), 

so that an increase in ?/i, or a decrease in /i, protracts the duration 
of a vibration. By a generalization of the language employed in 
the case of a material particle urged towards a position of equili- 
brium by a spring, m may be called the inertia of the system, and 



44.] DISSIPATIVE FORCES. 45 

/A the force of the equivalent spring. Thus an augmentation of 
mass, or a relaxation of spring, increases the periodic time. By 
means of this principle we may sometimes obtain limits for 
the value of a period, which cannot, or cannot easily, be calculated 
exactly. 

45. The absence of all forces of a frictional character is an 
ideal case, never realized but only approximated to in practice. 
The original energy of a vibration is always dissipated sooner or 
later by conversion into heat. But there is another source of loss, 
which though not, properly speaking, dissipative, yet produces 
results of m#Dh the same nature. Consider the case of a tuning- 
fork vibrating in vacuo. The internal friction will in time stop 
the motion, and the original energy will be transformed into 
heat. But now suppose that the fork is transferred to an open 
space. In strictness the fork and the air surrounding it consti- 
tute a single system, whose parts cannot be treated separately. 
In attempting, however, the exact solution of so complicated a 
problem, we should generally be stopped by mathematical diffi- 
culties, and in any case an approximate solution would be de- 
sirable. The effect of the air during a few periods is quite insig- 
nificant, and becomes important only by accumulation. We are 
thus led to consider its effect as a disturbance of the motion which 
would take place hi vacuo. The disturbing force is periodic (to 
the same approximation that the' vibrations are so), and may be 
divided into two parts, one proportional to the acceleration, and 
the other to the velocity. The former produces the same effect as 
an alt.eration in the mass of the fork, and we have nothing more 
to do with it at present. The latter is a force arithmetically pro- 
portional to the velocity, and always acts in opposition to the 
motion, and therefore produces effects of the same character as 
those due to friction. In many similar cases the loss of motion 
by communication may be treated under the same head as that 
due to dissipation proper, and is represented in the differential 
equation with a degree of approximation sufficient for acoustical 
purposes by a term proportional to the velocity. Thus 

u + /cu-hn*M = (1) 

is the equation of vibration for a system with one degree of 
freedom subject to frictional forces. The solution is 

M = ^e-i««cos{\V-i/c2. e-a} (2). 



46 ONE DEGREE OF FREEDOM. [45. 

If the friction be so great that ^k^ > n\ the solution changes its 
form, and no longer corresponds to an oscillatory motion ; but in 
all acoustical applications #c is a small quantity. Under these 
circumstances (2) may be regarded as expressing a harmonic 
vibration, whose amplitude is not constant, but diminishes in 
geometrical progression, when considered after equal intervals of 
time. The diflTerence of the logarithms of successive extreme 
excursions is nearly constant, and is called the Logarithmic Decre- 
ment. It is expressed by \tcT, if t be the periodic time. 

The frequency, depending on n* — \/c*, involves only the second 
power ot k; so that to the first order of approximation the friction 
has no effect on the pei^iod, — a principle of very general application. 

The vibration here considered is called the free vibration. It 
is that executed by the system, when disturbed from equilibrium, 
and then left to itself 

46. We must now turn our attention to another problem, not 
less important, — the behaviour of the system, when subjected to an 
external force varying as a harmonic frmction of the time. In 
order to save repetition, we may take at ouce the more general 
case including friction. If there be no friction, we have only to 
put in our results /c = 0. The differential equation is 

u+#fi + n*ii= Ecospt (1). 

Assume u == a cos (pt — e) (2), 

and substitute : 

a (n' — p") cos (p^ "~ €) — ^^ siJ^ (pt — c) 

= £^ cos € cos (pt — c) — Esine sin (pt — e) ; 

whence, on equating coefficients of cos (pt — c), sin (pt — c), 

a(««-^) = ^co8e| 

so that the solution may be written 

"=— r^co8(p^-€) (4), 

where tan 6 = -^—, (5). 

This is called a forced vibration; it is the response of the system 
to a force imposed upon it from without, and is maintained by the 
continued operation of that force. The amplitude is proportional 



46.] FORCED VIBRATIONS. 47 

to E — the magnitude of the force, and the period is the same 

as that of the force. 

Let us now suppose E given, and trace the effect on a given 

system of a variation in the period of the force. The effects 

produced in different cases are not strictly similar; because the 

frequency of the vibrations produced is always the same as that of 

the force, and therefore variable in the comparison which we are 

about to institute. We may, however, compare the energy of the 

system in different cases at the moment of passing through the 

position of equilibrium. It is necessary thus to specify the moment 

at which the energy is to be computed in each case, because the 

total energy is not invariable throughout the vibration. During 

one part of the period the system receives energy from the 

impressed force, and during the remainder of the period yields it 

back again. 

From (4), if w = 0, 

energy x li- x sin^ e, 

and is therefore a maximum, when sin € = 1, or, from (5),p = n. If 
the maximum kinetic energy be denoted by Tq, we have 

r=rosin«€ (6). 

The kinetic energy of the motion is therefore the greatest possible, 
when the period of the force is that in which the system would 
vibrate freely under the influence of its own elasticity (or other 
internal forces), without friction. The vibration is then by (4) 
and (5), 

M = - Sin nt : 

UK 

and, if k be small, its amplitude is very great. Its phase is a 

quarter of a period behind that of the force. 

The case, where p = 7i, may also be treated independently. 

Since the period of the actual vibration is the same as that 

natural to the system, 

u 4- n^u = 0, 

so that the differential equation (1) reduces to 

icu=^EGospty 
whence by integration 

u= - \ cosptdt= -- sin pt, 

K J TJ/C 

as before. 



48 ONE DEGREE OF FREEDOM. [46. 

If p be less than n, the retardation of phase relatively to the 
force lies between zero and a quarter period, and when p is greater 
than n, between a quarter period and a half period. 

In the case of a system devoid of friction, the solution is 

E 
w= J -.cospt (7). 

When p is smaller than n, the phase of the vibration agrees with 
that of the force, but when p is the greater, the sign of the vibra- 
tion is changed. The change of phase from complete agreement 
to complete disagreement, which is gradual when friction acts, 
here takes place abruptly as p passes through the value n. At the 
^ame time the expression for the amplitude becomes infinite. Of 
course this only means that, in the case of equal periods, friction 
must be taken into account, however small it may be, and however 
insignificant its result when p and n are not approximately equal. 
The limitation as to the magnitude of the vibration, to which we 
are all along subject, must also be borne in mind. 

That the excursion should be at its maximum in one direction 
while the generating force is at its maximum in the opposite 
direction, as happens, for example, in the canal theory of the tides, 
is sometimes considered a paradox. Any difficulty that may be 
felt will be removed by considering the extreme case, in which the 
^' spring " vanishes, so that the natural period is infinitely long. In 
fact we need only consider the force acting on the bob of a com- 
mon pendulum swinging freely, in which case the excursion on one 
side is greatest when the action of gravity is at its maximum 
in the opposite direction. When on the other hand the inertia of 
the system is very small, we have the other extreme case in which 
the so-called equilibrium theory becomes applicable, the force and 
excursion being in the same phase. 

When the period of the force is longer than the natural period, 
the effect of an increasing friction is to introduce a retardation 
in the phase of the displacement varying from zero up to a quarter 
period. If, however, the period of the natural vibration be the 
longer, the original retardation of half a period is diminished by 
something short of a quarter period ; or the effect of friction is to 
accelerate the phase of the displacement estimated from that corre- 
sponding* to the absence of friction. In either case the influence 
of friction is to cause an approximation to the state of things that 
would prevail if friction were paramount. 



46.] PRINCIPLE OF SUPERPOSITION. 49 

If a force of nearly equal period with the free vibrations 
vary slowly to a maximum and then slowly decrease, the dis- 
placement does not reach its maximum until after the force has 
begun to diminish. Under the operation of the force at its 
maximum, the vibration continues to increase until a certain limit 
is approached, and this increase continues for a time even although 
the force, having passed its maximum, begins to diminish. On 
this principle the retardation of spring tides behind the dajrs of 
new and full moon has been explained \ 

47. From the linearity of the equations it follows that the 
motion resulting from the simultaneous action of any number of 
forces is the simple sum of the motions due to the forces taken 
separately. £ach force causes the vibration proper to itself, 
without regard to the presence or absence of any others. The 
peculiarities of a force are thus in a manner transmitted into the 
motion of the system. For example, if the force be periodic in 
time T, so will be the resulting vibration. Elach harmonic element 
of the force will call forth a corresponding harmonic vibration 
in the system. But since the retardation of phase €, and the ratio 
of amplitudes a : E,is not the same for the different components, 
the resulting vibration, though periodic in the same time', is dif- 
ferent in character from the force. It may happen, for instance, 
that one of the components is isochronous, or nearly so, with the 
fi^e vibration, in which case it will manifest itself in the motion 
out of all proportion to its original importance. As another 
example we may consider the case of a system acted on by two 
forces of nearly equal period. The resulting vibration, being 
compounded of two nearly in unison, is intermittent, according to 
the principles explained in the last chapter. 

To the motions, which are the immediate effects of the im- 
pressed forces, must always be added the term expressing free 
vibrations, if it be desired to obtain the most general solution. 
Thus in the case of one impressed force, 

w = ^^^^cos(p^-€)4-^e-*««cos{Vn«-i/c».«-a} (1), ' 

where A and a are arbitrary. 

48. The distinction between /arced and free vibrations is very 
important, and must be clearly understood. The period of the 

^ Airy's Tides and Waves, Art. S2S. 
R. ^ 



50 ONE DEGREE OF FREEDOM. [48. 

former is determined solely by the force which is supposed to act 
on the system from without ; while that of the latter depends only 
on the constitution of the system itself. Another point of differ- 
ence is that so long as the external influence continues to operate, 
a forced vibration is permanent, being represented strictly by a 
harmonic function ; but a free vibration gradually dies away, be- 
coming negligible after a time. Suppose, for example, that the 
system is at rest when the force E cos pt begins to operate. Such 
finite values must be given to the constants A and a in (1) of § 47, 
that both u and u are initially zero. At first then there is a 
free vibration not less important than its rival, but after a time 
friction reduces it to insignificance, and the forced vibration is left 
in complete possession of the field. This condition of things will 
continue so long as the force operates. When the force is removed, 
there is, of course, no discontinuity in the values of u or u, but 
the forced vibration is at once converted into a free vibration, 
and the period of the force is exchanged for that natural to the 
system. 

During the coexistence of the two vibrations in the earlier part 
of the motion, the curious phenomenon of beats may occur, in 
case the two periods diflFer but slightly. For, n and p being nearly 
equal, and k small, the initial conditions are approximately satis- 
fied by 

u^a cos {pt — €) — ae~H cos {Vn* — ^^' . ^ — e]. 

There is thus a rise and fall in the motion, so long as e*"*** remains 
sensible. This intermittence is very conspicuous in the earlier 
stages of the motion of forks driven by electro-magnetism (§ 63), 
[and may be utilized when it is desired to adjust n and p to 
equality. The initial beats are to be made slower and slower, 
until they cease to be perceptible. The vibration then swells 
continuously to a maximum.] 

49. Vibrating systems of one degree of freedom may vary in 
two ways according to the values of the constants n and k. The 
distinction of pitch is sufficiently intelligible ; but it is worth while 
to examine more closely the consequences of a greater or less 
degree of damping. The most obvious is the more or less rapid 
extinction of a free vibration. The effect in this direction may be 
measured by the number of vibrations which must elapse before 
the amplitude is reduced in a given ratio. Initially the amplitude 
may be taken as unity ; after a time t, let it be 0. Initially = e~**^ 



49.] VABIOUS DBGREES OF DAMPING. 51 

2 

If t^XTf we have x — log 0. In a system subject to only a 

n'i 

moderate degree of damping, we may take approximately, 

T = 27r -r n ; 

80 that a; = — -logO (1). 

tcir 

This gives the number of vibrations which are performed, before 
the amplitude falls to 6, 

The influence of damping is also powerfully felt in a forced 
vibration, when there is a near approach to isochronism In the 
case of an exact equality between p and n, it is the damping alone 
which prevents the motion becoming infinite. We might easily 
anticipate that when the damping is small, a comparatively slight 
deviation from perfect isochronism would cause a large falling off 
in the magnitude of the vibration, but that with a larger damping 
the same precision of adjustment would not be required. From 
the equations 

r=ro8in«6, tan6 = -^, 

"-/-^J^- «^ 

so that if ic be small, p must be very nearly equal to w, in order to 
produce a motion not greatly less than the maximum. 

The two principal effects of damping may be compared by 
eliminating k between (1) and (2). The result is 



\oge_ ,p_n\ / T 



where the sign of the square root must be so chosen as to make 
the right-hand side negative. 

If, when a system vibrates freely, the amplitude be reduced in 
the ratio 6 after x vibrations ; then, when it is acted on by a force 
(/)), the energy of the resulting motion will be less than in the 
case of perfect isochronism in the ratio T : T^. It is a matter of 
indifference whether the forced or the free vibration be the higher; 
all depends on the interval. 

In most cases of interest the interval is small; and then, putting 
p^n-\-Sn, the formula may be written. 



lQg^_255n /~T~ 
X ~ n V To-T ^'' 



4—2 



52 



ONE DEGREE OF FREEDOM. 



[49. 



The following table calculated from these formulae has been 
given by Helmholtz^: 



Interval oorresponding to a redaction 
of the resonance to one-tenth. 



Number of vibrations after which the 
intensity of a firee vibration is re- 



VA »UV AVOVTAACaftAW W VftAV «OA« *A*. 

r:ro=i:io. 


daced to one-tenth. 


1 tone. 


38-00 


J tone. 


1900 


^ tone. 


9-50 


1 tone. 


6-33 


Whole tone. 


4-75 


^ tone. 


3-80 


^ tone = minor third. 


317 


{^ tone. 


2-71 


Two whole tones = major third. 


2-37 



Formula (4) shews that, when Bn is small, it varies coBteris 
paribus as -. 

60. From observations of forced vibrations due to known 
forces, the natural period and damping of a system may be deter- 
mined. The formulse are 

£^8in€ , ^ . 
cos(pt — €), 



t^ = 



where 



pK 
tan€= /^ ,. 



On the equilibrium theory we should have 

E 
ti = — cos pt. 

The ratio of the actual amplitude to this is 

J^sinc E n'sinc 
pK ■ n* pK ' 

If the equilibrium theory be known, the comparison of ampli- 
tudes tells us the value of 



n'sine 



pte 
n'sine 



say 



P^ 



= o, 



^ TonewqfJindMn^em, Srd edition, p. 881. 



50.] STRING WITH LOAD. 53 

and € is also known, whence 

, ^ /- co8€\ J ©sine .-. 

n» = o»-f- 1 ), and k— (1). 

\ a J a — C0S6 

61. As has been already stated, the distinction of forced and 
free vibrations is important ; but it may be remarked that most of 
the forced vibrations which we shall have to consider as affecting 
a system, take their origin ultimately in the motion of a second 
system, which influences the first, and is influenced by it. A 
vibration may thus have to be reckoned as forced in its relation 
to a system whose limits are fixed arbitrarily, even when that 
system has a share in determining the period of the force which 
acts upon it. On a wider view of the matter embracing both the 
systems, the vibration in question will be recognized as free. An 
example may make this clearer. A tuning-fork vibrating in air 
is part of a compound system including the air and itself, and 
in respect of this compound system the vibration is free. But 
although the fork is influenced by the reaction of the air, yet the 
amount of such influence is small. For practical purposes it is 
convenient to consider the motion of the fork as given, and that of 
the air as forced. No error will be committed if the actual motion 
of the fork (as influenced by its surroundings) be taken as the 
basis of calculation. But the peculiar advantage of this mode of 
conception is manifested in the case of an approximate solution 
being required. It may then suffice to substitute for the actual 
motion, what would be the motion of the fork in the absence of 
air, and afterwards introduce a correction, if necessary. 

62. Illustrations of the principles of this chapter may be 
drawn from all parts of Acoustics. We will give here a few 
applications which deserve an early place on account of their 
simplicity or importance. 

A string or wire ACB is stretched between two fixed points 
A and B, and at its centre carries a mass M, which is supposed to 
be so considerable as to render the mass of the string itself negli- 
gible. When M is pulled aside from its position of equilibrium, 
and then let go, it executes along the line CM vibrations, which 
are the subject of inquiry. AC — CB = a, CM = x. The tension 
of the string in the position of equilibrium depends on the amount 
of the stretching to which it has been subjected. In any other 



54 



OKE DEGREE OF FREEDOM. 



[52. 



position the tension is greater ; but we limit ourselves to the case 
of vibrations so small that the additional stretching is a negligible 
fraction of the whole. On this condition the tension may be 
treated as constant. We denote it by T. 




and 



IG.9. 



Thus, kinetic energy = \Md^, 



a? 



potential energy =27 {Va* + a:' - a} = T — approximately. 

The equation of motion (which may be derived also inde- 
pendently) is therefore 



a 



.(1). 



from which we infer that the mass M executes harmonic vibra- 
tions, whose period 

^ = 2--\/S <2)- 

The amplitude and phase depend of course on the initial cir- 
cumstances, being arbitrary so far as the differential equation is 
concerned. 

Equation (2) expresses the manner in which t varies with each 
of the independent quantities T, M,a: results which may all be 
obtained by consideration of the dimensions (in the technical sense) 
of the quantities involved. The argument from dimensions is so 
often of importance in Acoustics that it may be well to consider 
this first instance at length. 

In the first place we must assure ourselves that of all the 
quantities on which t may depend, the only ones involving a 



52.] METHOD OF DIMENSIONS. 55 

reference to the three fundamental units — of length, time, and 

mass — are a, M, and T, Let the solution of the problem be 

written 

r=f(a,M,T) ; (3). 

This equation must retain its form unchanged, whatever may 
be the fundamental units by means of which the four quantities 
are numerically expressed, as is evident, when it is considered 
that in deriving it no assumptions would be made as to the mag- 
nitudes of those units. Now of all the quantities on which / 
depends, T is the only one involving time ; and since its dimen- 
sions are (Mass) (Length) (Time)"*, it follows that when a and M 
are constant, t« T"*; otherwise a change in the unit of time 
would necessarily disturb the equation (3). This being admitted, 
it is easy to see that in order that (3) may be independent of the 
unit of length, we must have t x r~* . a*, when M is constant ; and 
finally, in order to secure independence of the unit of mass, 

Tocr-*.JIf*.a*. 

To determine these indices we might proceed thus : — assume 

TxT'.ifi'.a'; 

then by considering the dimensions in time, space, and mass, we 
obtain respectively 

1 = - 2a:, = a? + -?, = a: + y, 

whence as above a? = — i, y = i» z = \. 

There must be no mistake as to what this argument does and 
does not prove. We have assumed that there is a definite 
periodic time depending on no other quantities, having dimen- 
sions in space, time, and mass, than those above mentioned. For 
example, we have not proved that t is independent of the ampli- 
tude of vibration. That, so far as it is true at all, is a consequence 
of the linearity of the approximate differential equation. 

From the necessity of a complete enumeration of all the 
quantities on which the required result may depend, the method 
of dimensions is somewhat dangerous ; but when used with proper 
care it is unquestionably of great power and value. 

63. The solution of the present problem might be made the 
foundation of a method for the absolute measurement of pitch. 
The principal impediment to accuracy would probably be the 



56 



OSE VB^BXE OF FREEDOM. 



[53. 



dUBcnhy of making M mifBdeatiy kfge in relatioQ to the muB of 
the wire, tritboat at the Miiie time lowering the note too modi in 
th& miuneal ncale. 



<=r 



riQ.ta 



1' 



o 



The wire may be Htretched by a weight M' attached to its 
further eml beyond a bridge or pulley at B, The periodic time 
would be calculated from 



-^-V 



aM 
2gM' 



.(!)• 



The ratio of if ': Jf is given by the balance. If a be measured 
in feet, and g ■■ «32'2, the periodic time is expressed in seconds. 

64. In an ordinary musical string the weight, instead of being 
concentrated in the centre, is uniformly distributed over its length. 
Nevertholess the present problem gives some idea of the nature of 
the gravest vibration of huch a string. Let us compare the two 
ciutes more closely, supposing the amplitudes of vibration the same 
at the middle point. 




riG. II. 



Whon the uniform string is straight, at the moment of passing 
thriHigh the position of equilibrium, its different parts are ani- 
mated with a variable velocity, increasing from either end towards 



54.] COMPARISON WITH UNIFORM STRING. 57 

the centre. If we attribute to the whole mass the velocity of the 
centre, it is evident that the kinetic energy will be considerably 
over-estimated. Again, at the moment of maximum excursion, 
the uniform string is more stretched than its substitute, which 
follows the straight courses AMy MB, and accordingly the poten- 
tial energy is diminished by the substitution. The concentration 
of the mass at the middle point at once increases the kinetic 
energy when a? = 0, and decreases the potential energy when i? = 0, 
and therefore, according to the principle explained in § 44, prolongs 
the periodic time. For a string then the period is less than that 
calculated from the formula of the last section, on the supposition 
that M denotes the mass of the string. It will afterwards appear 
that in order to obtain a correct result we should have to take 
instead of M only (4/ir*)if. Of the factor 4/7r* by far the more 
important part, viz. ^, is due to the difference of the kinetic 
energies. 

55. As another example of a system possessing practically but 
one degree of freedom, let us consider the vibration of a spring, one 
end of which is clamped in a vice or otherwise held fast, while the 
other carries a heavy mass. 

In strictness, this system like the last has 
an iuiinite number of independent modes of vi- C j 

bration; but, when the mass of the spring is ^^ 

relatively small, that vibration which is nearly 
independent of its inertia becomes so much the Fioi2. 
most important that the others may be ignored. 
Pushing this idea to its limit, we may regard the 
spring merely as the origin of a force urging the 
attached mass towards the position of equilibrium, 
and, if a certain point be not exceeded, in simple '^ 
proportion to the displacement. The result is a ^ 
harmonic vibration, with a period dependent on 
the stiffness of the spring and the mass of the 
load. 

56. In consequence of the oscillation of the centre of inertia, 
there is a constant tendency towards the communication of motion 
to the supports, to resist which adequately the latter must be 
very firm and massive. In order to obviate this inconvenience. 




58 



ONE DEGREE OF FREEDOM. 



[56. 



O O 



two precisely similar springs and loads may be mounted on 

the same framework in a symmetrical manner. 

If the two loads perform vibrations of equal 

amplitude in such a manner that the motions 

are always opposite, or, as it may otherwise be 

expressed, with a phase-difference of half a 

period, the centre of inertia of the whole system 

remains at rest, and there is no tendency to set 

the framework into vibration. We shall see in a 

future chapter that this peculiar relation of phases 

will quickly establish itself, whatever may be the 

original disturbance. In fact, any part of the 

motion which does not conform to the condition 

of leaving the centre of inertia unmoved is soon 

extinguished by damping, unless indeed the 

supports of the system are more than usually 

firm. 



vvl 



O 



67. As in our first example we found a rough illustration of 
the fundamental vibration of a musical string, so here with the 
spring and attached load we may compare a uniform slip, or bar, 
of elastic material, one end of which is securely fastened, such for 
instance as the tongue of a reed instrument. It is true of course 
that the mass is not concentrated at one end, but distributed 
over the whole length ; yet on account of the smallness of the 
motion near the point of support, the inertia of that part of 
the bar is of but little account. We infer that the fundamental 
vibration of a uniform rod cannot be very different in character 
from that which we have been considering. Of course for pur- 
poses requiring precise calculation, the two systems are suflBciently 
distinct ; but where the object is to form clear ideas, pi'ecision may 
often be advantageously exchanged for simplicity. 

In the same spirit we may regard the combination of two 
springs and loads shewn in Fig. 13 as a representation of a 
tuning-fork. The instrument, which has been much improved 
of late years, is indispensable to the acoustical investigator. On 
a large scale and for rough purposes it may be made by welding 
a cross piece on the middle of a bar of steel, so as to form a T, and 
then bending the bar into the shape of a horse-shoe. On the 
handle a screw should be cut. But for the better class of tuning- 
forks it is preferable to shape the whole out of one piece of steel. 



57.] TUNING-FORKS. 59 

A division running from one end down the middle of a bar is first 
made, the two parts opened out to form the prongs of the fork, 
and the whole worked by the hammer and file into the required 
shape. The two prongs must be exactly symmetrical with respect 
to a plane passing through the axis of the handle, in order that 
during the vibration the centre of inertia may remain unmoved, 
— unmoved, that is, in the direction in which the prongs 
vibrate. 

The tuning is effected thus. To make the note higher, the 
equivalent inertia of the system must be reduced. This is done 
by filing away the ends of the prongs, either diminishing their 
thickness, or actually shortening them. On the other hand, to 
lower the pitch, the substance of the prongs near the bend may 
be reduced, the effect of which is to diminish the force of the 
spring, leaving the inertia practically unchanged ; or the inertia 
may be increased (a method which would be preferable for 
temporary purposes) by loading the ends of the prongs with 
wax, or other material. Large forks are sometimes provided with 
moveable weights, which slide along the prongs, and can be fixed 
in any position by screws. As these approach the ends (where the 
velocity is greatest) the equivalent inertia of the system increases. 
In this way a considerable range of pitch may be obtained fi'om 
one fork. The number of vibrations per second for any position 
of the weights may be marked on the prongs. 

The relation between the pitch and the size of tuning-forks is 
remarkably simple. In a future chapter it will be proved that, 
provided the material remains the same and the shape constant, 
the period of vibration varies directly as the linear dimension. 
Thus, if the linear dimensions of a tuning-fork be doubled, its 
note falls an octave. 

68. The note of a tuning-fork is a nearly pure tone. Imme- 
diately after a fork is struck, high tones may indeed be heard, 
corresponding to modes of vibration, whose nature will be subse- 
quently considered ; but these rapidly die away, and even while 
they exist, they do not blend with the proper tone of the fork, 
partly on account of their very high pitch, and partly because 
they do not belong to its harmonic scale. In the forks examined 
by Helmholtz the first of these overtones had a frequency from 5*8 
to 6*6 times that of the proper tone. 

Tuning-forks are now generally supplied with resonance cases, 



60 ONE DEGREE OF FREEDOM. [58. 

whose effect is greatly to augment the volume and purity of the 
sound, according to principles to be hereafter developed. In 
order to excite them, a violin or cello bow, well supplied with 
rosin, is drawn across the prongs in the direction of vibration. 
The sound so produced will last a minute or more. 

59. As standards of pitch tuning-forks are invaluable. The 
pitch of organ-pipes rapidly varies with the temperature and with 
the pressure of the wind ; that of strings with the tension, which 
can never be retained constant for long; but a tuning-fork kept 
clean and not subjected to violent changes of temperature or 
magnetization, preserves its pitch with great fidelity. 

[But it must not be supposed that the vibrations of a fork are 
altogether independent of temperature. According to the obser- 
vations of McLeod and Clarke^ the frequency falls by *00011 of its 
value for each degree Cent, of elevation.] 

By means of beats a standard tuning-fork may be copied with 
very great precision. The number of beats heard in a second is 
the difference of the frequencies of the two tones which produce 
them ; so that if the beats can be made so slow as to occupy half 
a minute each, the firequencies differ by only l-30th of a vibra- 
tion. Still greater precision might be obtained by Lissajous' 
optical method. 

Very slow beats being difficult of observation, in consequence 
of the uncertainty whether a falling off in the sound is due to 
interference or to the gradual dying away of the vibrations, 
Scheibler adopted a somewhat modified plan. He took a fork 
slightly different in pitch from the standard — whether higher or 
lower is not material, but we will say, lower, — and counted the 
number of beats, when they were sounded together. About four 
beats a second is the most suitable, and these may be counted for 
perhaps a minute. The fork to be adjusted is then made slightly 
higher than the auxiliary fork, and tuned to give with it precisely 
the same number of beats, as did the standard. In this way a 
copy as exact as possible is secured. To facilitate the counting 
of the beats Scheibler employed pendulums, whose periods of 
vibration could be adjusted. 

[The question between slow and quick beats depends upon the 
circumstances of the case. It seems to be sometimes supposed 
that quick beats have the advantage as admitting of greater 

1 Phil. Trans. ISSO, p. 12. 



59.] scheibleb's tonometer. 61 

relative accuracy of counting. But a little consideration shews 
that in a comparison of frequencies we are concerned not with the 
relative, but with the absolute accuracy of the counting. If we 
miscount the beats in a minute by one, it makes just the same 
error in the result, whether the whole number of beats be 60 or 
240. 

When the sounds are pure tones and are well maintained, it is 
advisable to use beats much slower than four per second. By 
choosing a suitable position it is often possible to make the 
intensities at the ear equal ; and then the phase of silence, 
corresponding to antagonism of equal and opposite, vibrations, is 
extremely well marked. Taking advantage of this we may deter- 
mine slow beats with very great accuracy by observing the time 
which elapses between recurrences of silence. In favourable cases 
the whole number of beats in the period of observation may be 
fixed to within one-tenth or one-twentieth of a single beat, a 
degree of accuracy which is out of the question when the beats 
are quick. In this way beats of periods exceeding 30 seconds may 
be utilised with excellent effect ^] 

60. The method of beats was also employed by Sch^ibler to 
determine the absolute pitch of his standards. Two forks were 
tuned to an octave, and a number of others prepared to bridge 
over the interval by steps so small that each fork gave with its 
immediate neighbours in the series a number of beats that could 
be easily counted. The difference of frequency corresponding to 
each step was observed with all possible accuracy. Their sum, 
being the difference of frequencies for the interval of the octave, 
was equal to the frequency of that fork which formed the starting 
point at the bottom of the series. The pitch of the other forks 
could be deduced. 

If consecutive forks give four beats per second, 65 in all will 
be required to bridge over the interval from c' (256) to c" (512), 
On this account the method is laborious ; but it is probably the 
most accurate for the original determination of pitch, as it is 
liable to no errors but such as care and repetition will eliminate. 
It may be observed that the essential thing is the measurement 
of the difference of frequencies for two notes, whose ratio of 
frequencies is independently known. If we could be sure of its 
accuracy, the interval of the fifth, fourth, or even major third, might 

1 Acoustical Obsenrations, Phil, Mag. May, 1SS2, p. 842. 



62 ONE DEGREE OF FREEDOM. [60. 

be substituted for the octave, with the advantage of reducing the 
number of the necessary interpolations. It is probable that with 
the aid of optical methods this course might be successfully 
adopted, as the corresponding Lissajous' figures are easily recog- 
nised, and their steadiness is a very severe test of the accuracy 
with which the ratio is attained. 

[It is essential to the success of this method that the pitch of 
eax^h of the numerous sounds employed should be definite, and in 
particular that the vibrations of any fork should take place at the 
same rate whether that fork be sounding in conjunction with its 
neighbour above or with its neighbour below. There is no reason 
to doubt that this condition is sufficiently satisfied in the case of 
independent tuning-forks; but an attempt to replace forks by a 
set of reeds, mounted side by side on a common wind-chest, has 
led to error, owing to a disturbance of pitch by mutual inter- 
action \] 

The frequency of large tuning-forks may be determined by 
allowing them to trace a harmonic curve on smoked paper, which 
may conveniently be mounted on the circumference of a revolving 
drum. The number of waves executed in a second of time gives 
the frequency. 

In many cases the use of intermittent illumination described 
in § 42 gives a convenient method of determining an unknown 
frequency. 

61. A series of forks ranging at small intervals over an octave 
is very useful for the determination of the frequency of any 
musical note, and is called Scheibler's Tonometer. It may also 
be used for tuning a note to any desired pitch. In either case 
the frequency of the note is determined by the number of beats 
which it gives with the forks, which lie nearest to it (on each 
side) in pitch. 

For tuning pianofortes or organs, a set of twelve forks may be 
used giving the notes of the chromatic scale on the equal tempe- 
rament, or any desired system. The corresponding notes are 
adjusted to unison, and the others tuned by octavea It is better, 
however, to prepare the forks so as to give four vibrations per 
second less than is above proposed. E^h note is then tuned a 
little higher than the corresponding fork, until they give when 
sounded together exactly four beats in the second. It will be 

1 Nature, xvii. pp. 12, 26 ; 1877. 



61. J scheibler's tonometer. 63 

observed that the addition (or subtraction) of a constant number 
to the frequencies is not the same thing as a mere displacement 
of the scale in absolute pitch. 

In the ordinary practice of tuners a! is taken from a fork, and 
the other notes determined by estimation of fifths. It will be 
remembered that twelve true fifths are slightly in excess of seven 
octaves, so that on the equal temperament system each fifth is a 
little flat. The tuner proceeds upwards from a! by successive 
fifths, coming down an octave after about every alternate step, in 
order to remain in nearly the same part of the scale. Twelve 
fifths should bring him back to a. If this be not the case, the 
work must be readjusted, until all the twelve fifths are too flat by, 
as nearly as can be judged, the same small amount. The in- 
evitable error is then impartially distributed, and rendered as little 
sensible, as possible. The octaves, of course, are all tuned true. 
The following numbers indicate the order in which the notes may 
be taken : 

a% b d c% df d't e' ff% g' g'% o! 0!% V d' d't d!' d"% e" 
13 5 16 8 19 11 3 14 6 17 9 1 12 4 16 7 18 10 2 

In practice the equal temperament is only approximately 
attained; but this is perhaps not of much consequence, considering 
that the system aimed at is itself by no means perfection. 

Violins and other instruments of that class are tuned by true 
fifths from a\ 

62. In illustration of forced vibration let us consider the case 
of a pendulum whose point of support is subject to a small hori- 
zontal harmonic motion. Q is the bob attached by a fine wire to 

a moveable point P. OP = a:„, p p 

PQ = l^ and x is the horizontal 
co-ordinate of Q. Since the 
vibrations are supposed small, 
the vertical motion may be 
neglected, and the tension of 
the wire equated to the weight 
of Q. Hence for the horizontal 

motion ^ + /ci + y (a? — aro) = 0. ar^ 1^ 

Now a?o oc cos pt ; so that putting g-i-l^n^ our equation takes 
the form already treated of, viz. 

x + KX-^ n^x = E cos pt 




64 



ONE DEGREE OF FREEDOM. 



[62. 



If p be equal to n, the motion is limited only by the friction. 
The assumed horizontal harmonic motion for P may be realized by 
means of a second pendulum of massive construction, which carries 
P with it in its motion. An efficient arrangement is shewn in 
the figure. A, B are iron rings screwed into a beam, or other firm 

/^/ C/5. 




support ; C, D similar rings attached to a stout bar, which carries 
equal heavy weights E, F, attached near its ends, and is supported 
in a horizontal position at right angles to the beam by a wire 
passing through the four rings in the manner shewn. When the 
pendulum is made to vibrate, a point in the rod midway between 
C and D executes a harmonic motion in a direction parallel to 
CD, and may be made the point of attachment of another pen- 
dulum PQ. If the weights E and F be very great in relation 
to Q, the upper pendulum swings very nearly in its own proper 
period, and induces in Q a forced vibration of the same period. 
When the length PQ is so adjusted that the natural periods of the 
two pendulums are nearly the same, Q will be thrown into violent 
motion, even though the vibration of P be of but inconsiderable 
amplitude. In this case the difference of phase is about a quarter 
of a period, by which amount the upper pendulum is in advance. 
If the two periods be very different, the vibrations either agree 
or are completely opposed in phase, according to equations (4) 
and (5) of § 46. 



63.] RELATION OF AMPLITUDE AND PHASE. 65 

63. A very good example of a forced vibration is aflforded by 
a fork under tke influence of an intermittent electric cuiTent, 

Fi G. /e. 




sa 



r 



3 



1 



, xir 



whose period is nearly equal to its own. ACB is the fork ; E a 
small electro-magnet, formed by winding insulated wire on an iron 
core of the shape shewn in E (similar to that known as 'Siemens' 
armature'), and supported between the prongs of the fork. When 
an intermittent current is sent through the wire, a periodic force 
acts upon the fork. This force is not expressible by a simple 
cirtjular function ; but may be expanded by Fourier's theorem in a 
series of such functions, having periods t, J t, ^ t, &c. If any of 
these, of not too small amplitude, be nearly isochronous with the 
fork, the latter will be caused to vibrate ; otherwise the effect is 
insignificant. In what follows we will suppose that it is the 
complete period t which nearly agrees with that of the fork, and 
consequently regard the series expressing the periodic force as 
reduced to its first term. 

In order to obtain the maximum vibration, the fork must be 
carefully tuned by a small sliding piece or by wax, until its natural 
period (without friction) is equal to that of the force. This is best 
done by actual trial. When the desired equality is approached, 
and the fork is allowed to start from rest, the force and com- 
plementary free vibration are of nearly equal amplitudes and 
frequencies, and therefore (§ 48) in the beginning of the motion 
produce heats^ whose slowness is a measure of the accuracy of 
the adjustment. It is not until after the free vibration has had 
time to subside, that the motion assumes its permanent character. 
The vibrations of a tuning-fork properly constructed and mounted 
are subject to very little damping; consequently a very slight 
deviation from perfect isochronism occasions a marked falling off 
in the intensity of the resonance. 

The amplitude of the forced vibration can be observed with 
sufficient accuracy by the ear or eye; but the experimental 



64 



ONE DEGREE OF FREEDOM. 



[62. 



If p be equal to n, the motion is limited only by the friction. 
The assumed horizontal harmonic motion for P may be realized by 
means of a second pendulum of massive construction, which carries 
P with it in its motion. An efficient arrangement is shewn in 
the figure. Ay B are iron rings screwed into a beam, or other firm 




support ; (7, D similar rings attached to a stout bar, which carries 
equal heavy weights E, F, attached near its ends, and is supported 
in a horizontal position at right angles to the beam by a wire 
passing through the four rings in the manner shewn. When the 
pendulum is made to vibrate, a point in the rod midway between 
C and D executes a harmonic motion in a direction parallel to 
CDy and may be made the point of attachment of another pen- 
dulum PQ. If the weights E and F be very great in relation 
to Q, the upper pendulum swings very nearly in its own proper 
period, and induces in Q a forced vibration of the same period. 
When the length PQ is so adjusted that the natural periods of the 
two pendulums are nearly the same, Q will be thrown into violent 
motion, even though the vibration of P be of but inconsiderable 
amplitude. In this case the difference of phase is about a quarter 
of a period, by which amount the upper pendulum is in advance. 
If the two periods be very different, the vibrations either agree 
or are completely opposed in phase, according to equations (4) 
and (5) of § 46. 



63.] 



RELATION OF AMPLITUDE AND PHASE. 



65 



63. A very good example of a forced vibration is aflforded by 
a fork under tke influence of an intermittent electric current, 




Ft c. le. 



^_^ 



o 

r 



^x 



whose period is nearly equal to its own. ACB is the fork ; E a 
small electro-magnet, formed by winding insulated wire on an iron 
core of the shape shewn in E (similar to that known as ' Siemens * 
armature'), and supported between the prongs of the fork. When 
an intermittent current is sent through the wire, a periodic force 
acts upon the fork. This force is not expressible by a simple 
circular function ; but may be expanded by Fourier's theorem in a 
series of such functions, having periods t,\t,\ t, &c. If any of 
these, of not too small amplitude, be nearly isochronous with the 
fork, the latter will be caused to vibrate ; otherwise the effect is 
insignificant. In what follows we will suppose that it is the 
complete period t which nearly agrees with that of the fork, and 
consequently regai*d the series expressing the periodic force as 
reduced to its first term. 

In order to obtain the maximum vibration, the fork must be 
carefully tuned by a small sliding piece or by wax, until its natural 
period (without friction) is equal to that of the force. This is best 
done by actual trial. When the desired equality is approached, 
and the fork is allowed to start from rest, the force and com- 
plementary free vibration are of nearly equal amplitudes and 
frequencies, and therefore (§ 48) in the beginning of the motion 
produce fceate, whose slowness is a measure of the accuracy of 
the adjustment. It is not until after the free vibration has had 
time to subside, that the motion assumes its permanent character. 
The vibrations of a tuning-fork properly constructed and mounted 
are subject to very little damping; consequently a very slight 
deviation from perfect isochronism occasions a marked falling off 
in the intensity of the resonance. 

The amplitude of the forced vibration can be observed with 
sufficient accuracy by the ear or eye; but the experimental 

R. "S 



66 ONE DEGREE OF FREEDOM. [63. 

verification of the relations pointed out by theory between its 
phase and that of the force which causes it, requires a modified 
arrangement. 

Two similar electro-magnets acting on similar forks, and in- 
cluded in the same circuit are excited by the same intermittent 
current. Under these circumstances it is clear that the systems 
will be thrown into similar vibrations, because they are acted on 
by equal forces. This similarity of vibrations refers both to phase 
and amplitude. Let us suppose now that the vibrations are 
effected in perpendicular directions, and by means of one of 
Lissajous' methods are optically compounded. The resulting figure 
is necessarily a straight line. Starting from the case in which, the 
amplitudes are a maximum, viz. when the natural periods of both 
forks are the same as that of the force, let one of them be put a 
little out of tune. It must be remembered that whatever their 
natural periods may be, the two forks vibrate in perfect unison 
with the force, and therefore with one another. The principal 
effect of the difference of the natural periods is to destroy the 
synchronism of phase. The straight line, which previously re- 
presented the compound vibration, becomes an ellipse, and this 
remains perfectly steady, so long as the forks are not touched. 
Originally the forks are both a quarter period behind the force. 
When the pitch of one is slightly lowered, it falls still more behind 
the force, and at the same time its amplitude diminishes. Let the 
difference of phase between the two forks be e', and the ratio of 
amplitudes of vibration a : a©. Then by (6) of § 46 

a = Uq cos e'. 

The following table shews the simultaneous values of a : a^ 
and e'. 

a : an c' 



1-0 





•9 


25<' 50' 


•8 


36« 52' 


•7 


45» 34' 


•6 


53» 7' 


•5 


60« 


•4 


66» 25' 


•3 


720 32' 


•2 


78« 27' 


•1 


840 15'' 



^ Tonempjindungent 8rd edition, p. 190. 



63.] FORK INTERRUPTER. 67 

It appears that a considerable alteration of phase in either 
direction may be obtained without very materially reducing the 
amplitude. When one fork is vibrating at its maximum, the 
other may be made to differ from it on either side by as much as 
60^ in phase, without losing more than half its amplitude, or by 
as much as 45^ without losing more than half its energy. By 
allowing one fork to vibrate 45' in advance, and the other 45® 
in arrear of the phase corresponding to the case of maximum 
resonance, we obtain a phase difference of 90® in conjunction with 
an equality of amplitudes. Lissajous' figure then becomes a circle. 

[An intennittent electric current may also be applied to 
regulate the speed of a revolving body. The phonic wheel, in- 
vented independently by M. La Cour and by the author of this 
work\ is of great service in acoustical investigations. It may take 
various forms; but the essential feature is the approximate 
closing of the magnetic circuit of an electro-magnet, fed with an 
intermittent current, by one or more soft iron armatures carried 
by the wheel and disposed symmetrically round the circumference. 
If in the revolution of the wheel the closest passage of the 
armature synchronises with the middle of the time of excitation, 
the electro-magnetic forces operating upon the armature during 
its advance and its retreat balance one another. If however the 
wheel be a little in arrear, the forces promoting advance gain an 
advantage over those hindering the retreat of the armature, and 
thus upon the whole the magnetic forces encourage the rotation. 
In like manner if the phase of the wheel be in advance of that 
first specified, forces are called into play which retard the motion. 
By a self-acting adjustment the rotation settles down into such 
a phase that the driving forces exactly balance the resistances. 
When the wheel runs lightly, and the electric appliances are 
moderately powerful, independent driving may not be needed. In 
this case of course the phase of closest passage must follow that 
which marks the middle of the time of magnetisation. If, as is 
sometimes advisable, there be an independent driving power, the 
phase of closest passage may either precede or follow that of 
magnetisation. 

In some cases the oscillations of the motion about the phase 
into which it should settle down are very persistent and interfere 
with the applications of the instrument. A remedy may be 
found in a ring containing water or mercury, revolving concen- 

1 Nature, May 23, 1878. 



68 0N£ DEGREE OF FREEDOM. [63. 

trically. When the rotation is uniform, the fluid revolves like a 
solid body and then exercises no influence. But when from any 
cause the speed changes, the fluid persists for a time in the former 
motion, and thus brings into play forces tending to damp out 
oscillations.] 

64. The intermittent current is best obtained by a fork- 
interrupter invented by Helmholtz. This may consist of a fork 
and electro-magnet mounted as before. The wires of the magnet 
are connected, one with one pole of the battery, and the other with 
a mercury cup. The other pole of the battery is connected with 
a second mercury cup. A U-shaped rider of insulated wire is 
carried by the lower prong just over the cups, at such a height 
that during the vibration the circuit is alternately made and 
broken by the passage of one end into and out of the mercury. 
The other end may be kept permanently immersed. By means 
of the periodic force thus obtained, the effect of friction is com- 
pensated, and the vibrations of the fork permanently maintained. 
In order to set another fork into forced vibration, its associated 
electro-magnet may be included, either in the same driving-circuit, 
or in a second, whose periodic interruption is effected by another 
rider dipping into mercury cups^ 

The modus operandi of this kind of self-acting instrument is 
often imperfectly apprehended. If the force acting on the fork 
depended only on its position — on whether the circuit were open 
or closed — the work done in passing through any position would 
be undone on the return, so that after a complete period there 
would be nothing outstanding by which the eflect of the frictional 
forces could be compensated. Any explanation which does not 
take account of the retardation of the current is wholly beside the 
mark. The causes of retardation are two : irregular contact, and 
self-induction. When the point of the rider first touches the 
mercury, the electric contact is imperfect, probably on account of 

^ I have arranged seyeral intermpters on the aboye plan, all the component 
parts being of home mana£actare. The forks were made by the village blacksmith. 
The oops consisted of iron thimbles, soldered on one end of copper slips, the 
farther end being screwed down on the base board of the instrument. Some 
means of adjusting the leyel of the mercory surface is necessary. In Helmholtz* 
interrupter a horse-shoe electro-magnet embracing the fork is adopted, but I am 
inclined to prefer the present arrangement, at any rate if the pitch be low. In 
some cases a greater motive power is obtained by a horse-shoe magnet acting on a 
soft iron armature carried horizontally by the upper prong and perpendicular to it. 
I have usually found a single Smee cell sufficient battery power. 



64.] FORK INTERRUPTER. 69 

adhering air. On the other hand, in leaving the mercury the 
contact is prolonged by the adhesion of the liquid in the cup to 
the amalgamated wire. On both accounts the current is retarded 
behind what would correspond to the mere position of the fork. 
But, even if the resistance of the circuit depended only on the 
position of the fork, the current would still be retarded by its self- 
induction. However perfect the contact may be, a finite current 
cannot be generated until afber the lapse of a finite time, any 
more than in ordinary mechanics a finite velocity can be suddenly 
impressed on an inert body. From whatever causes arising^ the 
effect of the retardation is that more work is gained by the fork 
during the retreat of the rider from the mercury, than is lost 
during its entrance, and thus a balance remains to be set off 
against friction. 

If the magnetic force depended only on the position of the fork, 
the phase of its first harmonic component might be considered to 
be ISO^ in advance of that of the fork s own vibration. The re- 
tardation spoken of reduces this advance. If the phase-difference 
be reduced to 90^ the force acts in the most favourable manner, 
and the greatest possible vibration is produced. 

It is important to notice that (except in the case just refeiTcd 
to) the actual pitch of the interrupter differs to some extent frx>m 
that natural to the fork according to the law expressed in (5) of 
§ 46, € being in the present case a prescribed phase-difference 
depending on the nature of the contacts and the magnitude of the 
self-induction. If the intermittent current be employed to drive 
a second fork, the maximum vibration is obtained, when the 
frequency of the fork coincides, not with the natural, but with the 
modified frequency of the interrupter. 

The deviation of a tuning-fork interrupter from its natural 
pitch is practically very small ; but the fact that such a deviation 
is possible, is at first sight rather surprising. The explanation (in 
the case of a small retardation of current) is, that during that half 
of the motion in which the prongs are the most separated, the 
electro-magnet acts in aid of the proper recovering power due to 
rigidity, and so naturally raises the pitch. Whatever the relation 
of phases may be, the force of the magnet may be divided into 

1 Any desired retardation might be obtained, in default of other means, by 
attaching the rider, not to the prong itself, bat to the further end of a light 
straight spring carried by the prong and set into forced yibration by the motion of 
its point of attachment. 



70 OXE DEGREE OF FREEDOM. [64. 

two parts respectively proportional to the velocity and displacement 
(or acceleration). To the first exclusively is due the sustaining 
power of the force, and to the second the alteration of pitch. 

66. The general phenomenon of resonance, though it cannot 
be exhaustively considered under the head of one degree of 
freedom, is in the main referable to the same general principles. 
When a forced vibration is excited in one part of a system, all 
the other parts are also influenced, a vibration of the same period 
being excited, whose amplitude depends on the constitution of the 
system considered as a whole. But it not unfrequently happens 
that interest centres on the vibration of an 'outljdng part whose 
connection with the rest of the system is but loose. In such a case 
the part in question, provided a certain limit of amplitude be 
not exceeded, is very much in the position of a system possessing 
one degree of freedom and acted on by a force, which may be 
regarded as given, independently of the natural period. The 
vibration is accordingly governed by the laws we have already 
investigated. In the case of approximate equality of periods to 
which the name of resonance is generally restricted, the ampli- 
tude may be very considerable, even though in other cases it 
might be so small as to be of little account; and the precision 
required in the adjustment of the periods in order to bring out 
the effect, depends on the degree of damping to which the system 
is subjected. 

Among bodies which resound without an extreme precision of 
tuning, may be mentioned stretched membranes, and strings asso- 
ciated with sounding-boards, as in the pianoforte and the violin. 
When the proper note is sounded in their neighbourhood, these 
bodies are caused to vibrate in a very perceptible manner. The 
experiment may be made by singing into a pianoforte the note 
given by any of its strings, having first raised the corresponding 
damper. Or if one of the strings belonging to any note be plucked 
(like a harp string) with the finger, its fellows will be set into 
vibration, as may immediately be proved by stopping the first. 

The phenomenon of resonance is, however, most striking in 
cases where a very accurate equality of periods is necessary in 
order to elicit the full effect Of this class tuning-forks, mounted 
on resonance boxes, are a conspicuous example. When the unison 
is perfect the vibration of one fork will be taken up by another 
across the width of a room, but the slightest deviation of pitch 



65.] RESONANCK 71 

is sufficient to render the phenomenon almost insensible. Forks 
of 256 vibrations per second are commonly used for the purpose, 
and it is found that a deviation from unison giving only one beat 
in a second makes all the difference. When the forks are well 
tuned and close together, the vibration may be transferred back- 
wards and forwards between them several times, by damping them 
alternately, with a touch of the finger. 

Illustrations of the powerful effects of isochronism must be 
within the experience of every one. They are often of importance 
in very different fields from any with which acoustics is concerned. 
For example, few things are more dangerous to a ship than to lie 
in the trough of the sea under the influence of waves whose period 
is nearly that of its own natural rolling. 

66 a. It has already (§ 30) been explained how the super- 
position of two vibrations of equal amplitude and of nearly equal 
frequency gives rise to a resultant in which the sound rises and 
falls in beats. If we represent the two components by cos 27r/ii^, 
cos 27rn^, the resultant is 

2cos7r(w, — 7i,)^.cos7r(7ii+n3)^ (1); 

and it may be regarded as a vibration of frequency J (ui + n,), and 
of amplitude 2 cos tt (wi — n^) t In passing through zero the 
amplitude changes sign, which is equivalent to a change of phase 
of 180°, if the amplitude be regarded as always positive. This 
change of phase is readily detected by measurement in drawings 
traced by machines for compounding vibrations, and it is a feature 
of great importance. If a force of this character act upon a system 
whose natural frequency is ^ (rii + n.j), the effect produced is com- 
paratively small. If the system start from rest, the successive 
impulses cooperate at first, but after a time the later impulses 
begin to destroy the effect of former ones. The greatest response 
would be given to forces of frequency 7ii and ?i^, and not to a force 
of frequency ^ (w^ + /ij). 

If, as in some experiments of Prof. A. M. Mayer ', an otherwise 
steady sound is rendered intermittent by the periodic interposition 
of an obstacle, a very different result is arrived at. In this case 
the phase is resumed after each silence without reversal. If a 
force of this character act upon an isochronous system, the effect 
is indeed less than if there were no intermittence ; but as all the 

* Phil, Mag, May. 1875. 



72 ONE DEGREE OF FREEDOM. [65 a. 

impulses operate in the same sense without any antagonism, the 

response is powerful. One kind of intermittent vibration or force 

is represented by 

2(1 +COS 27rmf ) cos 27mf (2), 

in which n is the frequency of the vibration, and m the frequency 
of intermittence ^ The amplitude is here always positive, and 
varies between the values and 4. By ordinary trigonometrical 
transformation (2) may be put in the form 

2 cas 27rn^ + cos 27r (n '\-vi)t + cos 27r (n — m) ^ (3); 

which shews that the intermittent vibration in question is equiva- 
lent to three simple vibrations of frequencies n, n + nif n — m. 
This is the explanation of the secondary sounds observed by 
Mayer. 

The form (2) is of course only a particular case. Another in 
which the intensity of the intermittent sound rises more suddenly 
to its maximum is given by 

^COQ* Trmt cos 27mt (4), 

which may be transformed into 

f cos 2Tmt + cos 27r (n + m) ^ + cos 27r (n — vi) t 

+ i cos 27r (n + 2jn) ^ + J cos 27r (n - 2m) t (5). 

There are here four secondary sounds, the frequencies of the 
two new ones differing twice as much as before from that of the 
primary sound. 

The theory of intermittent vibrations is well illustrated by 
electrically driven forks. A fork interrupter of frequency 128 
gave a periodic current, by the passage of which through an 
electro-magnet a second fork of like pitch could be excited. The 
action of this current on the second fork could be rendered inter- 
mittent by short-cireuiting the electro-magnet. This was eflfected 
by another interrupter of frequency 4, worked by an independent 
current from a Smee cell. To excite the main cuirent a Grove 
cell was employed. When the contact of the second interrupter 
was permanently broken, so that the main current passed con- 
tinuously through the electro-magnet, the fork was, of course, 
most powerfully affected when tuned to 128. Scarcely any 
response was observable when the pitch was changed to 124 or 
132. But if the second interrupter were allowed to operate, so as 

^ Cram Brown and Tait. Edin, Proc. June, 1S7S. Acoostioal Obserrations ii. 
Phil. Mag. April, 1S80. 



65 a.] 



INTERMITTENT VIBRATIONS. 



73 



to render the periodic current through the electro-magnet inter- 
mittent, then the fork would respond powerfully when tuned to 
124 or 132 as well as when tuned to 128, but not when tuned to 
intermediate pitches, such as 126 or 130. 

The operation of the intermittence in producing a sensitive- 
ness w*hich would not otherwise exist, is easily understood. When 
a fork of frequency 124 starts from rest under the influence of a 
force of frequency 128, the impulses cooperate at first, but after ^ 
of a second the new impulses begin to oppose the earlier ones. 
After ^ of a second, another series of impulses begins whose effect 
agrees with that of the first, and so on. Thus if all these impulses 
are allowed to act, the resultant effect is trifling ; but if every 
alternate series is stopped off, a large vibration accumulates. 

Fig. 16 a. 




The most general expression for a vibration of frequency n, 
whose amplitude and phase are slowly variable with a frequency 
m, is 



{Ao-^Ai cos 27rm^ -h A j cos 4nr?nt + A^ cos GttdU -h . . .) 
+ Bi sin 2'jrmt + B^ sin ^irint + -B, sin Gimit + . . .J 

f (7o + C^i cos 2'mnt -f C, cos 47rr?if + G^ cos Girmt + . . .) . 
( + -Disin 27rmt + Z), sin 47rm^ +-D3 sin Ginnt + . . . J 



cos 27rr<f 



sm Zimt 

...(6); 

and this applies both to the case of beats (e.g. if Ai only be finite) 
and to such intermittence as is produced by the interposition of 
an obstacle. The vibration in question is accordingly in all cases, 
equivalent to a combination of simple vibrations of frequencies 

n, n + m, n — m, n + 2m, n — 2m, &c. 
It may be well here to emphasise that a simple vibration 
implies infinite continuance, and does not admit of variations of 
phase or amplitude. To suppose, as is sometimes done in optical 
speculations, that a train of simple waves may begin at a given 
epoch, continue for a certain time involving it may be a large 
number of periods, and ultimately cease, is a contradiction in terms. 



74 ONE DEGREE OF FREEDOM. [66. 

66. The solution of the equation for free vibration, viz. 

u + /ici + w'a = (1). 

may be put into another form by expressing the arbitrary con- 
stants of integration A and a in terms of the initial values of u 
and u, which we may denote by u© and u^ We obtain at once 

u = 6-i«^ \uo — 7 — ^ y-a [cos n'^ + ^ sin n'tyh (2), 

where n' = V/i'' — \i<^. 

If there be no friction, /c = 0, and then 

. sinn^ , . .Qv 

u = Uo |-w«cosn^ (3). 

n 

These results may be employed to obtain the solution of the 

complete equation 

u + /cu + n^M = CT (4), 

where 17 is an explicit function of the time ; for from (2) we see 
that the effect at time ^ of a velocity hu communicated at time 
^'is 

n 

The eflFect of i/' is to generate in time dt* a velocity Udtf, whose 
result at time t will therefore be 

n ^ ^ 

and thus the solution of (4) will be 

u^^,j^e^-^'-^^smn'(t-t')Udt' (o). 

If there be no friction, we have simply 

u = ^f8inn(t-t')Udt' (6), 

nj 

U being the force at time t\ 

The lower limit of the integrals is so far arbitrary, but it will 
generally be convenient to make it zero. 

On this supposition u and u as given by (6) vanish, when 
^ = 0, and the complete solution is 

, , f. sinn'^ / ,,./«. ,\\ 
u = e-*«* i ito — } — h Wo ( cos n ^ + — > sm n ^ I > 



+ i^ fe-i-^f-^) sin n' (t - e') Udt' (7), 

n Jo 



66.] INSTABILITY. 75 

or if there be no friction 

sin Tit 1 r^ . 

w = Uo \-ii^coant-\-- I 8inn(^ — O Udt' (8). 

n njQ ^ 

When t is sufficiently great, the complementary terms tend to 
vanish on account of the factor e"**S and may then be omitted. 

66 a. In § 66 we have limited the discussion to the case of 
greatest acoustical importance, that is, we have supposed that nf 
is real, as happens when 7i^ is positive, and k not too great. But 
a more general treatment of the problem of free vibrations is not 
without interest. Whatever may be the values of n' and k, the 
solution of (1) § 66 may be expressed 

XL ^ Ae^^^ + B e*^^ (1), 

where /^, fi^ are the roots of 

/i*4-/c/i + n- = (2). 

The case already discussed is that in which the values of /i are 
imaginary. The motion is then oscillatory, with amplitude which 
decreases if k be positive, but increases if le be negative. 

But if n*, though positive, be less than ^/c*, or if n^ be negative, 
n' becomes imaginary, that is fi becomes real. The motion 
expressed by (1) is then non-oscillatoiy, and it depends upon the 
sign of /i whether it increases or diminishes with the time. From 
the solution of (2), viz. 

M = -i/c±Jv('t»-4H0 (3), 

it is evident that if n^ be positive (and less than \i^) the two 
values of /a are of the same sign, and that the sign is the opposite 
of that of K. Hence if k be positive, both terms in (1) diminish 
with the time, so that the system, however disturbed, subsides 
again into a state of rest. If, on the contrary, k be negative, the 
motion increases without limit. 

We have still to consider the case of n'* negative. The real 
values of fi are then of opposite signa It is possible so to start 
the system from a displaced position that it shall approach asymp- 
totically the condition of rest in the configuration of equilibrium ; 
but unless a special relation between displacement and velocity is 
satisfied, the motion tends to increase w^ithout limit. Under these 
circumstances the equilibrium must be regarded as unstable. In 
this sense stability requires that n' and k be both positive. 

A word may not be out of place as to the eflFect of an im- 



76 ONE DEGREE OF FREEDOM. [66 a. 

pressed force upon a statically unstable system. If in § 46 we 
suppose /c = 0, the solution (7) does not change its form merely 
because n^ becomes negative. The fact that a system is suscep- 
tible of purely periodic motion under the operation of an external 
periodic force is therefore no evidence of stability. 

67. For most acoustical purposes it is suflBcient to consider 
the vibrations of the systems, with which we may have to deal, 
as infinitely small, or rather as similar to infinitely small vibra- 
tion& This restriction is the foundation of the important laws 
of isochronism for free vibrations, and of persistence of period 
for forced vibrations. There are, however, phenomena of a sub- 
ordinate but not insignificant character, which depend essentially 
on the square and higher powers of the motion. We will therefore 
devote the remainder of this chapter to the discussion of the 
motion of a sjrstem of one degree of freedom, the motion not being 
so small that the squares and higher powers can be altogether 
neglected. 

The approximate expressions for the kinetic and potential 
energies will be of the form 

If the sum of T and V be diflFerentiated with respect to the 
time, we find as the equation of motion 

rn^ii + fj^u + miuii + ^miu^ + ^fJLiu^ = Impressed Force, 

which may be treated by the method of successive approximation. 
For the sake of simplicity we will take the case where mi = 0, 
a supposition in no way affecting the essence of the question. 
The inertia of the system is thus constant, while the force of 
restitution is a composite function of the displacement, partly pro- 
portional to the displacement itself and partly proportional to 
its square — cwjcordingly unsymmetrical with respect to the position 
of equilibrium. Thus for free vibrations our equation is of the 

form 

ii + n^u + au^^^O (1). 

with the approximate solution 

u^A cosnt (2), 

where A — ^the amplitude — is to be treated as a small quantity. 
Substituting the value of ti expressed by (2) in the last 

term, we find 

A^ 
it 4- n*?* = — a -5- (1 +COS 2nf), 



67.] TERMS OF THE SECOND ORDER, 11 

whence for a second approximation to the value of u 

u^Aco97it— «- + ^--cosznt (3); 

shewing that the proper tone (n) of the system is accompanied 

by its octave (2n), whose relative importance increases with the 

amplitude of vibration. A trained ear can generally perceive the 

octave in the sound of a tuning-fork caused to vibrate strongly by 

means of a bow, and with the aid of appliances, to be explained 

later, the existence of the octave may be made manifest to any 

one. By followiug the same method the approximation can 

be carried further; but we pass on now to the case of a system 

in which the recovering power is symmetrical with respect to 

the position of equilibrium. The equation of motion is then 

approximately 

u + ?i»tt + /8a» = (4), 

which may be underetood to refer to the vibrations of a heavy 
pendulum, or of a load carried at the end of a straight spring. 

If we take as a first approximation u = A cos nt, corresponding 
to /8 = 0, and substitute in the term multiplied by /8, we get 

.. , j3A^ „ , 3/8^» 

u -h n^u = — ^ cos fint — . cos nt, 

4 4 

Corresponding to the last term of this equation, we should 
obtain in the solution a term of the form ^sinn^, becoming 
greater without limit with t This, as in a parallel case in the 
Lunar Theory, indicates that our assumed first approximation 
is not really an approximation at all, or at least does not continue 
to be such. If, however, we take as our starting point u = A cos mt, 
with a suitable value for m, we shall find that the solution may 
be completed with the aid of periodic terms only. In fact it is 
evident beforehand that all we are entitled to assume is that the 
motion is approximately simple harmonic, with a period ap- 
proxiniately the same, as if /3 = 0. A very slight examination 
is sufficient to shew that the term varying as w^ not only may, 
but must affect the period. At the same time it is evident 
that a solution, in which the period is assumed wrongly, no 
matter by how little, must at length cease to represent the motion 
with any approach to accuracy. 

We take then for the approximate equation 

u-\-nh(,= — ~r — cosm^— -r- cos3m^ (5), 



78 ONE DEGREE OF FREEDOM. [67. 

of which the solution will be 

. ^ /8-4' cos3m^ .^^ 

M = 4cosmt+ ^ 9m' -„. (6). 

provided that m be taken so as to satisfy 

or m» = »« + ^' (7). 

The term in fi thus produces two effects. It alters the pitch 
of the fundamental vibration, and it introduces the twelfth as 
a necessary accompaniment. The alteration of pitch is in most 
cases exceedingly small — depending on the square of the amplitude, 
but it is not altogether insensible. Tuning-forks generally rise 
a little, though very little, in pitch as the vibration dies away. 
It may be remarked that the same slight dependence of pitch 
on amplitude occurs when the force of restitution is of the 
form n'M-haw^ as may be seen by continuing the approximation 
to the solution of (1) one step further than (3). The result in that 
case is 

-'="'-^' («)• 

The difference m' — n' is of the same order in ii in both cases ; 
but in one respect there is a distinction worth noting, namely, 
that in (8) m* is always less than w', while in (7) it depends on 
the sign of /8 whether its effect is to raise or lower the pitch. 
However, in most cases of the unsymmetrical class the change 
of pitch would depend partly on a term of the form av^ and 
partly on another of the form /8w', and then 

, , ha^A^ . 3/8i4« ,^,, 

^=^- 6h^ +-4- (^>- 

[In all cases where the period depends upon amplitude, it is 
necessarily an even function thereof, a change of sign in the ampli- 
tude being merely equivalent to an alteration in phase of 180°.] 

• 

68. We now pass to the consideration of the vibrations 
forced on an unsymmetrical system by two harmonic forces 

Ecospt, F cos (qt — e). 

^ [A correction is here introduced, the necessity for which was pointed oat to me 
by Dr Burton.] 



68.J TERMS OF THE SECOND ORDER. 79 

The equation of motion is 

il-\-n^u^ -au^ + Ecospt-h F cos{qt- e) (1). 

To find a first approximation we neglect the term containing 

a. Thus 

u = e cos pt+/cos(qt — e) (2), 

where « = n^' f^n^-f <^>- 

Substituting this in the term multiplied by a, we get 

u 4- n'a = ^cos pt + i^cos {qt — £) 

- a ^-^ + 2 ^^^ ^^^ + 4- cos 2 {qt - c) + e/cos {0> - ?) « + e} 

+ efcos[{p'¥q)t-€\ , 
whence as a second approximation for u 

u = 6cosp«+/cos (}«-€) ^2^^ "2(7i*- V)^^^^ 

The additional terms represent vibrations having frequencies 
which are severally the doubles and the sum and diflFerence of 
those of the primaries. Of the two latter the amplitudes are 
proportional to the product of the original amplitudes, shewing 
that the derived tones increase in relative importance with 
the intensity of their parent tones. 

68a. If an isolated vibrating system be subject to internal 
dissipative influences, the vibrations cannot be permanent, since 
they are dependent upon an initial store of energy which suffers 
gradual exhaustion. In order that the motion may be maintained, 
the vibrating body must be in connection with a source of energy. 
We have already considered cases of this kind under the head of 
forced vibrations, where the system is subject to forces whose 
amplitude and phase are prescribed, independently of the be- 
haviour of the system. Such forces may have their origin in 
revolving mechanism (such as electric alternators) governed so as 
to move at a uniform speed. But more frequently the forces 
under consideration depend upon the vibrations of other systems. 



80 ONE DEGREE OF FREEDOM. [68 a. 

and then the question as to how the vibrations are to be main- 
tained represents itself. A good example is afforded by the case 
already discussed (§§ 63, 65) of a fork maintained in vibration 
electrically by means of currents governed by a fork interrupter. 
It has been pointed out that the performance of the latter 
depends upon the magnetic forces operative upon it differing in 
phase from the vibrations of the fork itself. With the interrupter 
may be classed for the present purpose almost all acoustical and 
musical instruments capable of providing a sustained sound. It 
may suflSce to mention vibrations maintained by wind (organ- 
pipes, harmonium reeds, seolian harps, &c.), by heat (singing 
flames, Rijke's tubes, &c.), by friction (violin strings, finger- 
glasses), and the slower vibrations of clock pendulums and watch 
balance-wheels. 

In considering whether proposed forces are of the right kind 
for the maintenance or encouragement of a vibration, it is often 
convenient to regard them as reduced to impulses. Suppose, to 
take a simple case, that a small horizontal positive impulse acts 
upon the bob of a vibrating pendulum. The effect depends, of 
course, upon the phase of the vibration at the instant of the 
impulse. If the bob be moving positively at the instant in 
question the vibration is encouraged, and this effect is a maximum 
when the positive motion is greatest, that is, when the impulse 
occurs at the moment of positive movement through the position 
of equilibrium. This is the condition of things aimed at in 
designing a clock escapement, for the effect of the force is then a 
maximum, in encouraging the vibration, and a minimum (zero to 
the first order of approximation) in disturbing the period. Of 
course, if the impulse be half a period earlier or later than is 
above supposed, the effect is to discourage the vibration, again 
without altering the period. In like manner we see that if the 
impulse occur at a moment of maximum elongation the effect is 
concentrated upon the period, the vibration being neither en- 
couraged nor discouraged. 

In most cases the force acting upon a vibrating system in 
virtue of its connection with a source of energy may be regarded 
as harmonic. It may then be divided into two parts, one pro- 
portional to the displacement u (or to the acceleration it), the 
second proportional to the velocity u. The inclusion of such 
forces does not alter the form of the equation of vibration 

u + icu + n»u = (1). 



68 a.] MAINTENANCE OF VIBRATIONS. 81 

By the first part (proportional to u) the pitch is modified, and by 
the second the coefficient of decay. If the altered k be still 
positive, vibrations gradually die down ; but if the eflFect of the 
included forces be to render k negative, vibrations tend on the 
contrary to increase. The only case in which according to (1) a 
steady vibration is possible, is when the complete value of k is 
zero. If this condition be satisfied, a vibration of any amplitude 
is permanently maintained. 

When K is negative, so that small vibrations tend to increase, 
a point is of course soon reached beyond which the approximate 
equations cease to be applicable. We may form an idea of the 
state of things which then arises by adding to equation (1) a 
term proportional to a higher power of the velocity. Let us take 

w+/cu+ic'ii» + n«a = (2), 

in which k and k are supposed to be small quantities. The 
approximate solution of (2) is 

u = A sin nt-\-—^ cos Znt (3), 

in which A is given by 

/c + 3acV^« = (4). 

From (4) we see that no steady vibration is possible unless k and 
K have opposite signs. If k and k be both positive, the vibration 
in all cases dies down ; while if k and k be both negative, the 
vibration (according to (2)) increases without limit. If /v be 
negative and k' positive, the vibration becomes steady and 
assumes the amplitude determined by (4). A smaller vibration 
increases up to this point, and a larger vibration falls down to it. 
If on the other hand k be positive, while k is negative, the steady 
vibration abstractedly possible is unstable, a departure in either 
direction from the amplitude given by (4) tending always to 
increase *. 

68 6. We will now consider briefly another and a very curious 
kind of maintenance, of which the peculiarity is that the maintain- 
ing influence operates with a frequency which is the double of 
that of the vibration maintained. Probably the best known 
example is that form of Melde s experiment, in which a fine string 
is maintained in transverse vibration by connecting one of its 
extremities with the vibrating prong of a massive tuning-fork, 

^ On Maintained Vibrations, Phil. Mag,, April, ISSS. 
R. 6 



82 ONE DEGREE OF FREEDOM. [68 6. 

the direction of motion of the point of attachment being parallel to 
the lefigth of the string. The effect of the motion is to render 
the tension of the string periodically variable ; and at first sight 
there is nothing to cause the string to depart from its equilibrium 
condition of straightness. It is known, however, that under these 
circumstances the equilibrium may become unstable, and that the 
string may settle down into a state of permanent and vigorous 
vibration, whose period is the double of that of the fork. 

As a simpler example, with but one degree of freedom, we 
may take a pendulum, formed of a bar of soft iron and vibrating 
upon knife-edges. Underneath is placed symmetrically a vertical 
bar electro-magnet, through which is caused to pass an electric 
current rendered intermittent by an interrupter whose frequency 
is twice that of the pendulum. The magnetic force does not tend 
to displace the pendulum from its equilibrium position, but 
produces the same sort of effect as if gravity were subject to a 
periodic variation of intensity. 

A similar result is obtained by causing the point of support 
of the pendulum to vibrate in a vertical path. If we denote this 
motion by 17 = /8 sin 2p^, the effect is as if gravity were variable by 
the term 4p*/8 sin 2pt 

Of the same nature are the crispations observed by Faraday* 
and others upon the surface of water which oscillates vertically. 
Faraday arrived experimentally at the conclusion that there were 
two complete vibrations of the support for each complete vibra- 
tion of the liquid. 

In the following investigation', relative to the case of one 
degree of freedom, we shall start with the assumption that a 
steady vibration is in progress, and inquire under what conditions 
the assumed state of things is possible. 

If the force of restitution, or " spring," of a body susceptible 
of vibration be subject to an imposed periodic variation, the 
differential equation takes the form 

16 -h /cu -h (n« - 2a sin 2pe) M = (1), 

in which le and a are supposed to be small. A similar equation 
would apply approximately to the case of a periodic variation in 
the effective mass of the body. The motion expressed by the 
solution of (1) can be regular only when it keeps perfect time 

* Phil. Trans, 1831, p. 299. 
2 phii^ jjjf^g^ ^prii^ ige3. 



68 &.] MAINTENANCE OF VIBRATIONS. 83 

with the imposed variations. It will appear that the necessary 
conditions cannot be satisfied rigorously by any simple harmonic 
vibration, but we may assume 

u * Ai empt 4- B^ cos pt 

+ Az sin Spt-hBtiCOS Spt+ A^ sin bpt + (2), 

in which, it is not necessary to provide for sines and cosines of even 
multiples of pt If the assumption be justifiable, the solution in 
(2) must be convergent. Substituting in the differential equation, 
and equating to zero the coefficients of sin pt, cos pt, &c. we find 

^1 {^^-P') - f^P^i - olBi + olBs = 0, 

Bi (n^-p^) + fcpAi - ol4i - olAj = ; 

Az (n:' - V) - SKpB^ - olBi + olBj = 0, 

^3 ( »^ - V) + 3 /c/)^ + aid 1 - a-d 5 = ; 

J 5 (/I* - 2op^) - oKpB, - aB^ + olB^ = 0, 

B, (n^ - 25j[)0 + OKpA, + ail, - a^^ = ; 

These equations shew that -^3, ^3 are of the order a relatively 
to Aif Bi\ that -4.5, B^ are of order a relatively to -4,, -B,, and 
so on. If we omit -4,, B^ in the first pair of equations, we find 
as a first approximation, 

A,{n^^p^)^{Kp-¥a)B,^0, 

whence ^ - - "^ = "^^^^ = ^^''r.M (3) 

Whence J,-^^ + a W^-jt)' V(«+icp) ^ ^' 

and (n«-p7 = a^-Acy (4). 

Thus, if a be given, the value of p necessary for a regular 
motion is definite ; and ;; having this value, the regular motion is 

w = P sin {pt + e), 

in which e, being equal to tan~* (BJAi), is also definite. On the 
other hand, as is evident at once from the linearity of the original 
equation, there is nothing to limit the amplitude of vibration. 

These characteristics are preserved however far it may be 
necessary to pursue the approximation. If A.i,n+it -Bam+i may be 
neglected, the first m pairs of equations determine the ratios of all 
the coefficients, leaving the absolute magnitude open; and they 
provide further an equation connecting p and a, by which the 
pitch is determined. 



84 . ONE DEGREE OF FREEDOM. [68 b. 

For the second approximation the second pair of equations 
give 



{■ 



whence 

aP 

i^ = P8in(p^ + €) + _-^^ cos (3;>< -h e) (5), 

and from the first pair 

tan€ = |n^-;)»-^'jj^j-(a + /c/>) (6), 

while p is determined by 

i"'-^'-„'-"y=«'-*'^^ (7). 

Returning to the first approximation, we see from (4) that the 
solution is possible only under the condition that a be not less 
than Kp. If a = xp, then p =^7i; that is, the imposed variation 
in the "spring" must be exactly twice as quick as the natural 
vibration of the body would be in the absence of friction. From 
(3) it appears that in this case e^O, which indicates that the 
spring is a minimum one-eighth of a period after the body has 
passed its position of equilibrium, and a maximum one-eighth of a 
period be/ore such passage. Under these circumstances the 
greatest possible amount of energy is communicated to the 
system ; and in the case contemplated it is just sufficient to 
balance the loss by dissipation, the adjustment being evidently 
independent of the amplitude. 

If a< xp sufficient energy cannot pass to maintain the motion, 
whatever may be the phase-relation ; but it a> tcp, the balance 
between energy supplied and energy dissipated may be attained 
by such an alteration of phase as shall diminish the former 
quantity to the required amount. The alteration of phase may 
for this purpose be indifferently in either direction ; but if e be 
positive, we must have 

pa = n« - V(a' - t^p^) ; 
while if 6 be negative 

p^ ^ n} + V(a' - f^P^y 

If a be very much greater than /(cp, e = ± ^-ir, which indicates 
that when the system passes through its position of equilibrium 
the spring is at its maximum or at its minimum. 

The inference from the equation that the adjustment of pitch 



68 6.] ABSOLUTE PITCH. 85 

must be absolutely rigorous for steady vibration will be subject to 
some modification in practice; otherwise the experiment could 
not succeed. In most cases v? is to a certain extent a function of 
amplitude; so that if v? have very nearly the required value, 
complete coincidence is attainable by the assumption of an 
amplitude of large and determinate amount without other 
alterations in the conditions of the system. 

The reader who wishes to pursue this subject is referred to a 
paper by the Author " On the Maintenance of Vibrations by Forces 
of Double Frequency, and on the Propagation of Waves through a 
Medium endowed with a Periodic Structure,"* in which the analysis 
of Mr Hill'' is applied to the present problem. 

68 c. The determination of absolute pitch by means of the 
siren has already been alluded to (§ 17). In all probability first- 
rate results might be got by this method if proper provision, with 
the aid of a phonic wheel for example, were made for uniform 
speed. In recent years several experimenters have obtained excel- 
lent results by various methods ; but a brief notice of these is all 
that our limits will allow. 

One of the most direct determinations is that of Koenig', to 
whom the scientific world has long been indebted for the construc- 
tion of much excellent apparatus. This depends upon a special 
instrument, consisting of a fork of 64 complete vibrations per 
second, the motion being maintained by a clock movement acting 
upon an escapement. A dial is provided marking ordinary time, 
and serves to record the number of vibrations executed. The 
performance of the fork is tested by a comparison between the 
instrument and any chronometer known to be keeping good time. 
The standard fork of 256 complete vibrations was compared with 
that of the instrument by observing the Lissajous*s figure appro- 
priate to the double octave. 

M. Koenig has also investigated the influence of resonators 
upon the pitch of forks. Thus without a resonator a fork of 256 
complete vibrations sounded in a satisfactory manner for about 90 
seconds. A resonator of adjustable pitch was then brought into 
proximity, and the pitch, originally much graver than that of the 

» P*a Mag,, August, 1887. 

^ On the Part of the Motion of the Lunar Perigree which is a Function of the 
Mean Motions of the Sun and Moon, Acta Mathematica 8 ; 1, 1886. Mr HilFs 
work was first published in 1877. 

3 Wied. Ann, ix. p. 394, 1880. 



86 ONE DEGREE OF FREEDOM. [68 C. 

fork, was gradually raised. Even when the resonator was still a 
minor third below the fork, there was observed a slight diminution 
in the duration of the vibratory movement, and at the same time 
an augmentation in the frequency of about '005. As the natural 
note of the resonator approached nearer to that of the fork, this 
diminution in the time and this increase in frequency became 
more pronounced up to the immediate neighbourhood of unison ; 
but at the moment when unison was established, the alteration of 
pitch suddenly disappeared, and the frequency became exactly the 
same as in the absence of the resonator. At the same time the 
sound was powerfully reinforced; but this exaggerated intensity 
fell off rapidly and the vibration died away after 8 or 10 seconds. 
The pitch of the resonator being again raised a little, the sound of 
the fork began to change in the opposite direction, being now as 
much too grave as before the unison was reached it had been too 
acute. The displacement then fell away by degrees, as the pitch 
of the resonator was fuii;her raised, and the duration of the 
vibrations gradually recovered its original value of about 90 
seconds. The maximum disturbance in the frequency observed 
by Koenig was "OSS complete vibrations. For the explanation 
of these effects see § 117. 

The temperature coeflBcient found by Koenig is '000112, so that 
the pitch of a 256 fork falls 0286 for each degree Cent, by which 
the temperature rises. 

In determinations of absolute pitch ^ by the Author of this work 
an electrically maintained interrupter fork, whose frequency may 
for example be 32, was employed to drive a dependent fork of 
pitch 128. When the apparatus is in good order, there is a fixed 
relation between the two frequencies, the one being precisely 
four times the other. The higher is of course readily compared 
by beats, or by optical methods, with a standard of 128, whose 
accuracy is to be Wsted. It remains to determine the frequency 
of the interrupter fork itself. 

For this pui-pose the interrupter is compared with the pendulum 
of a standard clock whose rate is known. The comparison may be 
direct, or the intervention of a phonic wheel (§ 63) may be invoked. 
In either case the pendulum of the clock is provided with a silvered 
bead upon which is concentrated the light from a lamp. Im- 
mediately in front of the pendulum is placed a screen perforated 
by a somewhat narrow vertical slit. The bright point of light 

^ Nature, xvii. p. 13, 1S77 ; PhiL Tram. 1SS3, Part I. p. 816. 



68 C.J ABSOLUTE PITCH. 87 

reflected by the bead is seen intermittently, either by looking over 
the prong of the interrupter or through a hole in the disc of the 
phonic wheel. In the first case there are 32 views per second, but 
in the latter this number is reduced by the intervention of the 
wheel. In the experiments referred to the wheel was so 
arranged that one revolution corresponded to four complete vibra- 
tions of the interrupter, and there were thus 8 views of the pen- 
dulum per second, instead of 32. Any deviation of the period of 
the pendulum from a precise multiple of the period of intermittence 
shews itself as a cycle of changes in the appearance of the flash 
of light, and an observation of the duration of this cycle gives the 
data for a pi*ecise comparison of frequencies. 

The calculation of the results is very simple. Supposing in 
the first instance that the clock is correct, let a be the number of 
cycles per second (perhaps ^) between the wheel and the clock. 
Since the period of a cycle is the time required for the wheel to 
gain, or lose, one revolution upon the clock, the frequency of revo- 
lution is 8 ± a. The frequency of the auxiliary fork is precisely 16 
times as great, i.e. 128 ± 16a. If 6 be the number of beats per 
second between the auxiliary fork and the standard, the frequency 

of the latter is 

128 ± 16a ± 6. 

An error in the mean rate of the clock is readily allowed for ; 
but care is required to ascertain that the actual rate at the time 
of observation does not differ appreciably from the mean rate. 
To be quite safe it would be necessarj' to repeat the deter- 
minations at intervals over the whole time required to rate the 
clock by observation of the stars. In this case it would probably 
be coDvenient to attach a counting apparatus to the phonic wheel. 

In the method of M'Leod and Clarke* time, given by a clock, 
is recorded automatically upon the revolving drum of a chrono- 
graph, which is maintained by a suitable governor in uniform 
rotation. The circumference of the drum is marked with a grating 
of equidistant lines parallel to the axis, and the comparison between 
the drum and the standard fork is effected by observation of the 
wavy pattern seen when the revolving grating is looked at past 
the edges of the vibrating prongs. These observers made a special 
investigation as to the effect of bowing a fork upon previously 
existing vibrations. Their conclusion is that in the case of un- 
loaded forks no sensible change of phase occurs. 

1 PhiL Tram, 1880, Part I. p. 1. 



88 ONE DEGREE OF FREEDOM. [68 C. 

In the chronographic method of Prof. A. M. Mayer^ the fork 
under investigation is armed with a triaDgular fragment of thin 
sheet metal, one milligram in weight, and actually traces its 
vibrations as a curve of sines upon smoked paper. The time is 
recorded by small electric discharges from an induction apparatus, 
under the control of a clock, and delivered from the same tmcing 
point. Although the disturbance due to the tracing point appears 
to be very small, it is doubtful whether this method could compete 
in respect of accuracy with those above described where the com- 
parison with the standard is optical or acoustical. On the other 
hand, it has the advantage of not requiring a uniform rotation of 
the drum, and the apparatus lends itself with facility to the deter- 
mination of small intervals of time after the manner originally 
proposed by T. Young^ 

68d. The methods hitherto described for the determination of 
absolute pitch, with the exception of that of Scheibler, may be 
regarded as rather mechanical in their character, and they depend 
for the most part upon somewhat special apparatus It is possible, 
however, to determine pitch with fair accuracy with no other 
appliances than a common harmonium and a watch, and as the 
process is instructive in respect of the theory of overtones, a short 
account will here be given of it*. 

The fundamental principle is that the absolute frequencies of 
two musical notes can be deduced from the interval between 
them, i.e. the ratio of their frequencies, and the number of beats 
which they occasion in a given time when sounded together. 
For example, if x and y denote the frequencies of two notes whose 
interval is an equal temperament major third, we know that 
y = 1*25992 x. At the same time the number of beats heard in a 
second depending upon the deviation of the third from true 
intonation, is 4y — ox. In the case of the notes of a harmonium, 
which are rich in overtones, these beats are readily counted, and 
thus two equations are obtained from which the values of x and y 
are at once found. 

Of course in practice the truth of an equal temperament third 
could not be taken for granted, but the diflBculty thence arising 
would be easily met by including in the counting all the three 

^ National Academy of Sciences, Washington, Memoirs, VoL ui. p. 43, 1884. 

• Lectures f Vol. i. p. 191. 

* Nature, Jan. 28, 1879. 



68 d] ABSOLUTE PITCH. 89 

major thirds which together make up an octave. Suppose, for 
example, that the frequencies of c, e, gt, c are respectively x, y, z, 
2x, and that the beats per second between x and y are a, between 
y and z are 6, and between z and 2x are c. Then 

4y — Oil; = a, 4fZ — oy^ 6, 8a: — oz = c, 
from which a? = ^ (25a + 206 + 16c), 

y = J (32a + 256 + 20c), 
z = J (40a + 326 + 25c). 

In the above statements the octave c — c is for simplicity 
supposed to be true. The actual error could readily be allowed 
for if required ; but in practice it is not necessary to use c at all, 
inasmuch as the third set of beats can be counted equally well 
between gf and c. 

The principal objection to the method in the above form is 
that it presupposes the absolute constancy of the notes, for 
example, that y is the same whether it is being sounded in 
conjunction with x or in conjunction with z. This condition is 
very imperfectly satisfied by the notes of a harmonium. 

In order to apply the fundamental principle with success, it is 
necessary to be able to check the accuracy of the interval which is 
supposed to be known, at the same time that the beats are being 
counted. If the interval be a major tone (9 : 8), its exactness is 
proved by the absence of beats between the ninth component of 
the lower and the eighth of the higher note, and a counting 
of the beats between the tenth component of the lower and the 
ninth of the higher note completes the necessary data for de- 
termining the absolute pitch. 

The equal temperament whole tone (1'12246) is intermediate 
between the minor tone (llllll) and the major tone (1*12500), 
but lies much nearer to the latter. Regarded as a disturbed 
major tone, it gives slow beats, and regai-ded as a disturbed 
minor tone it gives quick ones. Both sets of beats can be heard 
at the same time, and when counted (by two observers) give the 
means of calculating the absolute pitch of both notes. If x and y 
be the frequencies of the two notes, a and 6 the frequencies of the 
slow and quick beats respectively, 

9a?— 8y = a, 9y— 10a: = 6, 

whence a: = 9a + 86, y = 10a + 96. 

The application of this method in no way assumes the truth of 



90 ONE DEGREE OF FREEDOM. [68 d. 

the equal temperament whole tone, and in fact it is advantageous 
to flatten the interval somewhat, so as to make it lie more nearly 
midway between the major and the minor tone. In this way the 
rapidity of the quicker beats is diminished, which facilitates the 
counting. 

The course of an experiment is then as follows. The notes C 
and D are sounded together, and at a given signal the observers 
begin counting the beats situated at about d" and e' on the scale. 
After the expiration of a measured interval of time a second signal 
is given, and the number of both sets of beats is recorded. 

For further details of the method reference must be made to 
the original memoir, but one example of the results may be given 
here. The period being 10 minutes, the number of beats recorded 
were 2392 and 2341, giving x = 6709 as the pitch of C. 



CHAPTER IV. 

VIBRATING SYSTEMS IN GENERAL. 

69. We have now examined in some detail the oscillations 
of a system possessed of one degree of freedom, and the results, 
at which we have arrived, have a very wide application. But 
material systems enjoy in general more than one degree of 
freedom. In order to define their configuration at any moment 
several independent variable quantities must be specified, which, 
by a generalization of language originally employed for a point, 
are called the co-ordinates of the system, the number of indepen- 
dent co-ordinates being the index of freedom. Strictly speaking, 
the displacements possible to a natural system are infinitely 
various, and cannot be represented as made up of a finite number 
of displacements of specified type. To the elementary parts of 
a solid body any arbitrary displacements may be given, subject 
to conditions of continuity. It is only by a process of abstraction 
of the kind so constantly practised in Natural Philosophy, that 
solids are treated as rigid, fluids as incompressible, and other sim- 
plifications introduced so that the position of a system comes to 
depend on a finite number of co-ordinates. It is not, however, 
our intention to exclude the consideration of systems possessing 
infinitely various freedom; on the contrary, some of the most 
interesting applications of the results of this chapter will lie in 
that direction. But such systems are most conveniently conceived 
as limits of others, whose freedom is of a more restricted kind. 
We shall accordingly commence with a system, whose position 
is specified by a finite number of independent co-ordinates ^,, 
-^2, V^s* &c. 

70. The main problem of Acoustics consists in the investi- 
gation of the vibrations of a system about a position of stable 
equilibrium, but it will be convenient to commence with the 



92 



VIBRATING SYSTEMS IN GENERAL. 



[70. 



statical part of the subject. By the Principle of Virtual Velocities, 
if we. reckon the co-ordinates -^i, ^.j, Ac. from the configuration 
of equilibrium, the potential energy of any other configuration 
mil be a homogeneous quadratic function of the co-ordinates, 
provided that the displacement be suflBciently small. This quan- 
tity is called F, and represents the work that may be gained in 
passing from the actual to the equilibrium configuration. We may 
write 

Since by supposition the equilibrium is thoroughly stable, the 
quantities Cn, c^, Cu, &c. must be such that V is positive for all 
real values of the co-ordinates. 

71. If the system be displaced from the zero configuration 
by the action of given forces, the new configuration may be 
found from the Principle of Virtual Velocities. If the work done 
by the given forces on the hypothetical displacement S-^i, S^„ 
&c. be 

'^M^^'^H^'^ (1), 

this expression must be equivalent to SF, so that since S^j, S-^a, 
&c. are independent, the new position of equilibrium is deter- 
mined by 

Jr=*'. #.-*-«"^ <=i). 

or by (1) of §70, 

Cai^i + CaaVra -f Cas^s + =^a^ (3), 



where there is no distinction in value between Cr% and c,^. 

From these equations the co-ordinates may be determined in 
terms of the forcea If V be the determinant 



Cfl, Caa, Caa, 
^31* ^82* Caa, 



the solution of (3) may be written 

_ ^ dV dV 

(vCai dCaa 



(4), 



.(5). 



71.] RECIPROCAL RELATION. 93 

These equations determine -^i, -^a, &c. uniquely, since V does 
not vanish, as appears from the consideration that the equations 
dV/dy^i = 0, &c. could otherwise be satisfied by finite values of the 
co-ordinates, provided only that the ratios were suitable, which is 
contrary to the hypothesis that the system is thoroughly stable in 
the zero configuration. 

72. If -^1, ... S?i, ... and -^Z, ... S?/, ... be two sets of dis- 
placements and corresponding forces, we have the following re- 
ciprocal relation, 

as may be seen by substituting the values of the forces, when each 
side of (1) takes the form, 

Suppose in (1) that all the forces vanish except ^j and ^/; 
then 

^,^; = ^i>i (2). 

If the forces ^a and "Vi be of the same kind, we may suppose 
them equal, and we then recognise that a force of any type acting 
alone produces a displacement of a second type equal to the 
displacement of the first tjrpe due to the action of an equal force 
of the second type. For example, if A and B be two points 
of a rod supported horizontally in any manner, the vertical de- 
flection at A, when a weight W is attached at B, is the same as 
the deflection at B, when W is applied at A \ 

73. Since F is a homogeneous quadratic function of the co- 
ordinates, 

dV dV 

^""-ii*-*^*'* <»• 

or, if ^1, "^j, &c. be the forces necessary to maintain the dis- 
placement represented by V^ijV^ai&c* 

2r=^iti + *.^» + (2). 

If -^i + A-^^i, -^j + A-^/rg, &c. represent another displacement 
for which the necessary forces are'^i + AS?i, '^s + AS^jj&c, the 

> On this sabjeot, see Phil. Mag., Dec., 1874, and Maroh, 1875. 



94 VIBRATING SYSTEMS IN GENERAL. [73. 

corresponding potential energy is given by 

= 27 + *i Ai/ri + >P,A^, + ... 

SO that we may write 

2A7=r2^.A^ + 2A*.^4-2A^.A^ (3), 

where AFis the diflTerence of the potential energies in the two 

cases, and we must particularly notice that by the reciprocal 

relation, § 72 (1), 

2^.AVr = 2A*.Vr (4). 

From (3) and (4) we may deduce two important theorems, 
relating to the value of V for a system subjected to given dis- 
placements, and to given forces respectively. 

74. The first theorem is to the effect that, if given displace- 
ments (not sufficient by themselves to determine the configuration) 
be produced in a system by forces of corresponding types, the re- 
sulting value of F for the system so displaced, and in equilibrium, 
is as small as it can be under the given displacement conditions ; 
and that the value of F for any other configuration exceeds this 
by the potential energy of the configuration which is the difference 
of the two. The only difficulty in the above statement consists 
in understanding what is meant by * forces of corresponding types.* 
Suppose, for example, that the system is a stretched string, of 
which a given point P is to be subject to an obligatory displace- 
ment; the force of corresponding type is here a force applied 
at the point P itself. And generally, the forces, by which the 
proposed displacement is to be made, must be such as would do 
no work on the system, provided only that that displacement were 
not made. 

By a suitable choice of co-ordinates, the given displacement 
conditions may be expressed by ascribing given values to the first 
r co-ordinates '^i, V^a, ••• '^r, and the conditions as to the forces 
, will then be represented by making the forces of the remaining 
types S?r+i, "^r+ai &c. vanish. If -^ -h A-^/r refer to any other con- 
figuration of the system, and "V -h A*^ be the corresponding forces, 
we are to suppose that A-^i, A-^,, &c. as far as A-^^. all vanish. 
Thus for the first r suffixes A*^ vanishes, and for the remaining 



74.] STATICAL THEOREMS. 95 

suffixes ^ vanishes. Accordingly S^.A*^ is zero, and therefore 
SA^.*^ is also zero. Hence 

2AF=x2A>P.AVr (1), 

which proves that if the given displacements be made in any 
other than the prescribed way, the potential energy is increased 
by the energy of the diflTerence of the configurations. 

By means of this theorem we may trace the effect on V of any 
relaxation in the stiffness of a system, subject to given displacement 
conditions. For, if after the alteration in stiffness the original equi- 
librium configuration be considered, the value of V corresponding 
thereto is by supposition less than before ; and, as we have just 
seen, there will be a still further diminution in the value of V 
when the system passes to equilibrium under the altered con- 
ditions. Hence we conclude that a diminution in F as a function 
of the co-ordinates entails also a diminution in the actual value of 
V when a system is subjected to given displacements. It will 
be understood that in particular cases the diminution spoken of 
may vanish ^ 

For example, if a point P of a bar clamped at both ends be 
displaced laterally to a given small amount by a force there ap- 
plied, the potential energy of the deformation will be diminished 
by any relaxation (however local) in the stiffness of the bar. 

76. The second theorem relates to a system displaced hy given 
forceSf and asserts that in this case the value of V in equilibrium 
is greater than it would be in any other configuration in which 
the system could be maintained at rest under the given forces, by 
the operation of mere constraints. We will shew that the removal 
of constraints increases the value of F. 

The co-ordinates may be so chosen that the conditions of con- 
straint are expressed by 

^1 = 0, ^, = 0, ^, = (1). 

We have then to prove that when S^^+i, S?,.+„ &c. are given, the 
value of F is least when the conditions (1) hold. The second 
configuration being denoted as before by -^^i-fA-^/rj &c., we see 
that for suffixes up to r inclusive -^ vanishes, and for higher 
suffixes AS[^ vanishes. Hence 

^ See a paper on General Theorems relating to Equilibriom and Initial and 
Steady Motions. PhiL Mag., March, 1875. 



96 VIBRATING SYSTEMS IN GENERAL. [75. 

and therefore 

2dF=2A>P.A^ (2), 

shewing that the increase in V due to the removal of the con- 
straints is equal to the potential energy of the difference of the two 
configurations. 

76. We now pass to the investigation of the initial motion of 
a system which starts from rest under the operation of given 
impulsea The motion thus acquired is independent of any 
potential energy which the system may possess when actually 
displaced, since by the nature of impulses we have to do only 
with the initial configuration itself. The initial motion is also 
independent of any forces of a finite kind, whether impressed on 
the s}«tem from without, or of the nature of viscosity. 

If P, Q, -R be the component impulses, parallel to the axes, on 
a particle m whose rectangular co-ordinates are x, y, z, we have by 
D'Alembert's Principle 

lm(xSx + ySy + zSz)=^X(PBx + Q8j/ + RSz) (1), 

where i, y, z denote the velocities acquired by the particle in virtue 
of the impulses, and Bx, By, Bz correspond to any arbitrary dis- 
placement of the system which does not violate the connection of 
its parts. It is required to transform (1) into an equation expressed 
by the independent generalized co-ordinates. 

c/ 
For the first side, '/ 

. J , ■v ( . dx . dv , . dz\ 



-'^■If*'*-!!/ «• 

where T, the kinetic energy of the system is supposed to be 
expressed as a function of -^j, -^j, &c. 



76.] IMPULSES. 97 

On the second side, 

= f.S^^. + f,Sf,+ (3), 

The transformed equation is therefore 

(lR-f')»*'^(^|.-f-) «*■ + •■-» '*>• 

where S-^^i, S-^^,, &c. are now completely independent. Hence to 
determine the motion we have 

,-r=fi» y; =fa,&c (5), 

where fi, fa, &c. may be considered as the generalized components 
of impulse. 

77. Since T is a homogeneous quadratic function of the gene- 
ralized co-ordinates, we may take 

whence 



Ci= ,; = ttii-^/r, + ai2>|ra + aisY^s + 

^9= ,-7- = Oai-^l + a22^a + 023^3 + 

aya 



(2), 



/ 



where there is no distinction in value between Ors and a^ 
Again, by the nature of T, 

22'='^,-5T +^'lf + = f.^i + f.^.+ (3)- 

The theory of initial motion is closely analogous to that of the 
displacement of a system from a configuration of stable equilibrium 
by steadily applied forces. In the present theory the initial kinetic 
energy T bears to the velocities and impulses the same relations 
as in the former V bears to the displacements and forces respect- 
ively. In one respect the theory of initial motions is the more 
complete, inasmuch as T is exactly, while F is in general only 
approximately, a homogeneous quadratic function of the variables. 

K. 7 



98 VIBRATING SYSTEMS IN GENERAL. [77- 

If -^1, -^2, . . ., fi» ^2 . • • • denote oae set of velocities and impulses 
for a system started from rest, and -^Z, yjt^, ..., f/, fj', ... a second 
set, we may prove, as in § 72, the following reciprocal relation : 

fi'ti+f.'t.+.-.=fiti'+f«t/+ (4y. 

This theorem admits of interesting application to fluid motion. 
It is known, and will be proved later in the course of this work, 
that the motion of a frictionless incompressible liquid, which 
starts from rest, is of such a kind that its component velocities 
at any point are the corresponding differential coeflScients of a 
certain function, called the velocity-potential. Let the fluid be 
set in motion by a prescribed arbitrary deformation of the surface 
S o{ a, closed space described within it. The resulting motion is 
determined by the normal velocities of the elements of 8, which, 
being shared by the fluid in contact with them, are denoted by 
du/dn, if u be the velocity-potential, which interpreted physically 
denotes the impulsive pressure. Hence by the theorem, if v be 
the velocity-potential of a second motion, corresponding to 
another set of arbitrary surface velocities dv/dn, 



lhPn^'lh> «. 



— an equation immediately following from Green's theorem, if 
besides S there be only fixed solids immersed in the fluid. The 
present method enables us to attribute to it a much higher gene- 
rality. For example, the immersed solids, instead of being fixed, 
may be free, altogether or in part, to take the motion imposed 
upon them by the fluid pressures. 

78. A particular case of the general theorem is worthy of 
special notice. In the first motion let 

yjti^A, -^2 = 0, ^3 = ^4 = ^5 =0; 

and in the second, 

^/=o, ^2'=^, ?;=?;=?/ =0. 

Then f/ = f2 (1). 

In words, if, by means of a suitable impulse of the correspond- 
ing type, a given arbitrary velocity of one co-ordinate be impressed 
on a system, the impulse corresponding to a second co-ordinate 
necessary in order to prevent it from changing, is the same as 
would be required for the first co-ordinate, if the given velocity 
were impressed on the second. 

1 Thomaon and Tait, § 318 (/). 



78.] Kelvin's theorem. 99 

As a simple example, take the ease of two spheres A and B 
immersed in a liquid, whose centres are free to move along certain 
lines. If il be set in motion with a given velocity, B will 
naturally begin to move also. The theorem asserts that the 
impulse required to prevent the motion of B, is the same as if 
the functions of A and B were exchanged : and this even though 
there be other rigid bodies, (7, Z>, &c., in the fluid, either fixed, or 
free in whole or in part. 

The case of electric currents mutually influencing each other by 
induction is precisely similar. Let there be two circuits A and B, 
in the neighbourhood of which there may be any number of other 
wire circuits or solid conductors. If a unit current be suddenly 
developed in the circuit A, the electromotive impulse induced in 
B is the same as there would have been in A, had the current been 
forcibly developed in B, 

79. The motion of a system, on which given arbitrary velocities 
are impressed by means of the necessary impulses of the corre- 
sponding types, possesses a remarkable property discovered by 
Thomson. The conditions are that -^i, -^2, "^j, ..."^r are given, 
while fr+i, fr+2, ••• vanish. Let -^i, -^-,...^1, fj, &c. correspond to 
the actual motion ; and 

to another motion satisfying the same velocity conditions. For 
each suffix either A-^ or f vanishes. Now for the kinetic energy 
of the supposed motion, 

2(r+Ar)=(f, + Af)(^, + A^0 + ... 

= 2r+fiA^, + f,A^2+... 

+ Afx.i^i + Af3.i|r,4-... + AfiA^i+Af5Ai^,+ .... 
But by the reciprocal relation (4) of § 77 

of which the former by hypothesis is zero ; so that 

2Ar=AfiA^i + Af,A^2-f (1), 

shewing that the energy of the supposed motion exceeds that of 
the actual motion by the energy of that motion which would have 
to be compounded with the latter to produce the former. The 
motion actually induced in the system has thus less energy than 
any other satisfying the same velocity conditions. In a subsequent 
chapter we shall make use of this property to find a superior limit 
to the energy of a system set in motion with prescribed velocities. 



100 VIBRATING SYSTEMS IN GENERAL. [79. 

If any diminution be made in the inertia of any of the parts 
of a system, the motion corresponding to prescribed velocity 
conditions will in general undergo a change. The value of T will 
necessarily be less than before ; for there would be a decrease even 
if the motion remained unchanged, and therefore a fortiori when 
the motion is such as to make T an absolute minimum. Con- 
versely any increase in the inertia increases the initial value of T. 

This theorem is analogous to that of § 74. The analogue for 
initial motions of the theorem of § 75, relating to the potential 
energy of a system displaced by given forces, is that of Bertrand, 
and may be thus stated : — If a system start from rest under the 
operation of given impulses, the kinetic energy of the actual motion 
exceeds that of any other motion which the system might have 
been guided to take with the assistance of mere constraints, by the 
kinetic energy of the difference of the motions^ 

[The theorems of Kelvin and Bertrand represent different 
aspects of the same truth. Let us suppose that the prescribed 
impulse is entirely of the first type fi. Then T^^fi*^!, whether 
the motion be free or be subjected to any constraint. Further, 
under any given circumstances as to consti-aint, -^i is proportional 
to fi, and the ratio fi : '^i may be regarded as the moment of 
inertia ; so that 

Kelvin's theorem asserts that the introduction of a constraint 
can only increase the value of T when ^^ is given. Hence whether 
•^1 be given or not, the constraint can only increase the ratio of 
27 to '^^ or of fi to '^^. Both theorems are included in the 
statement that the moment of inertia is increased by the intro- 
duction of a constraint.] 

80. We will not dwell at any greater length on the mechanics 
of a system subject to impulses, but pass on to investigate 
Lagrange 8 equations for continuous motion. We shall suppose 
that the connections binding together the parts of the system 
are not explicit functions of the time; such cases of forced 
motion as we shall have to consider will be specially shewn to 
be within the scope of the investigation. 

By D'Alembert 8 Principle in combination with that of Virtual 
Velocities, 

2m {ithx + yhy 4- zhz) = 2 {Xix + Yhy + Zhz) (1), 

1 Thomson and Tait, § 311. Phil Mag. March, 1875. 



80.] Lagrange's equations, 101 

where Sx, Sy, Sz denote a displacement of the system of the most 
general kind possible without violating the connections of its 
parts. Since the displacements of the individual particles of 
the system are mutually related, Sx,,.. are not independent. The 
object now is to transform to other variables -^i, yjr^,.,., which 
shall be independent. We have 



xix = -J- (xSx) — J&c* , 



so that 



2m (xSx-\- yiy + Hhz) — -ri- 2m {xtx 4- yhy -\-zhz) — iT, 
But (§ 76) we have already found that 

2m (i&r + yiy + zhz) = —r- Byfr^ + --r- S^j 4- ..., 

while sr=|^s^, + ^^s^,+ ..., 

if T be expressed as a quadratic function of -^i, -^^Tj, ..., whose 
coefficients are in general functions of -^/ri, -^j,.... Also 

d (dT ,, \ d fdT\ ^, dT ^; 

inasmuch as -r. 8*^1 — ^;n'^i^ 

Accordingly 



+ 



(ife)-^w,+ (2). 



Thus, if the transformation of the second side of (1) be 

2(Z8ar+FSy + irS«) = ^,S^, + ^,8^, + (3), 

we have equations of motion of the form 

lf^).^ = >p (4). 

dt^dyfr' dir ^ 

Since 'VSylr denotes the work done on the system during a 
displacement 8^, ^ may be regarded as the generalized com- 
ponent of force. 

In the case of a conservative system it is convenient to 
separate from '9 those parts which depend only on the configura- 



102 VIBRATING SYSTEMS IN GENERAL. [80. 

tion of the system. Thus, if V denote the potential energy, we 
may write 

dAdyfrJ dyfr dylr ^^ ^' 

where ^ is now limited to the forces acting on the system which 
are not already taken account of in the term dVjd^. 

81. There is also another group of forces whose existence 
it is often advantageous to recognize specially, namely those 
arising from friction or viscosity. If we suppose that each 
particle of the system is retarded by forces proportional to its 
component velocities, the effect will be shewn in the fundamental 
equation (1) § 80 by the addition to the left-hand member of 
the terms 

where /««, Ky, k^ are coefficients independent of the velocities, 
but possibly dependent on the configuration of the system. The 
transformation to the independent co-ordinates '^i, '^s, &c. is 
effected in a similar manner to that of 

^111 (xhx -^ySy + z Bz) 

considered above (§ 80), and gives 

JT ^^' + rr ^»+ <i>' 

dYi dYi 

where F=\^ {kJ? + K^y^ + /c^«) 

= iftu^i' + i6a^,« + ... + 5i,^i^, + 5«^,ifr, 4- (2). 

F, it will be observed, is like T a homogeneous quadratic 
function of the velocities, positive for all real values of the 
variables. It represents half the rate at which energy is dissipated. 

The above investigation refers to retarding forces proportional 
to the absolute velocities ; but it is equally important to consider 
such as depend on the relative velocities of the parts of the 
system, and fortunately this can be done without any increase 
of complication. For example, if a force act on the particle x^ 
proportional to (x^ — i^), there will be at the same moment an 
equal and opposite force acting on the particle x^. The additional 
terms in the fundamental equation will be of the form 

which may be written 

««(*! - ^s) 8(a^ -a:j) = 8^1 rr {i*« (^1 ~ ^a)*} + ••• ' 



81.] THE DISSIPATION FUNCTION. 103 

and so on for any number of pairs of mutually influencing 
particles. The only effect is the addition of new terms to F, 
which still appears in the form (2)\ We shall see presently that 
the existence of the function F, which may be called the Dis- 
sipation Function, implies certain relations among the coefficients 
of the generalized equations of vibration, which carry with them 
important consequences'. 

The equations of motion may now be written in the form 

d /dT\ dT dF dV ' 

dt\dyir) dylr dyjr dylr ^ ^' 

82. We may now introduce the condition that the motion 
takes place in the immediate neighbourhood of a configuration 
of thoroughly stable equilibrium ; T and F are then homogeneous 
quadratic functions of the velocities with coefficients which are 
to be treated as constant, and F is a similar function of the 
co-ordinates themselves, provided that (as we suppose to be 
the case) the origin of each co-ordinate is taken to correspond 
with the configuration of equilibrium. Moreover all three 
functions are essentially positive. Since terms of the form dT/dyjr 
are of the second order of small quantities, the equations of motion 
become linear, assuming the form 

dAd^/ d^^d^ ^ ^' 

where under ^ are to be included all forces acting on the system 
not already provided for by the differential coefficients of F and V. 
The three quadratic functions will be expressed as follows : — 

i^ = i6ii^i' + i6«^2*+ ... +6„t,^,+ ... \ (2), 

where the coefficients a, 6, c are constants. 

From equation (1) we may of course fall back on previous 
results by supposing F and F, or F and T, to vanish. 

A third set of theorems of interest in the application to Elec- 

^ The differences referred to in (he text may of coarse pass into differential 
coefficients in the case of a body continuonsly deformed. 

^ The Dissipation Function appears for the first time, so far as I am aware, in 
a paper on General Theorems relating to Vibrations, published in the Proceedingt 
of the Mathematical Society for June, 1873. 



104 



VIBRATING SYSTEMS IN GENERAL, 



[82. 



tricity may be obtained by omitting T and F, while F is retained, 
but it is unnecessary to pursue the subject here. 

If we substitute the values of T, F and F, and write D for djdt, 
we obtain a system of equations which may be put into the form 

.^21-^1 + e^'^^ + e»'^j + ... = ^s 



.(3). 



where e^ denotes the quadratic operator 

ert^anD^-^hrgD-JfCn (4). 

It must be particularly remarked that since 

it follows that ^n^'^^t (5). 

[The theory of motional forces, i.e. forces proportional to the 
velocities, has been further developed in the second edition of 
Thomson and Tait*s Natural Philosophy (1879). In the most 
general case the equations may be written 



iti^)'^^^''^'''^'''^^^^^ 



«+...=>Fil 



d/dT\ dV 



(6), 



where br,=^b„, firs=fiir (7). 

Of these the terms with the coefficients 6 can be derived from 
the dissipation function 

The terms in /8 on the other hand do not represent dissipation, 
and are called the gyrostatic terms. 

If we multiply the first of equations (6) by -^^i, the second by 
'^s, &c., and then add, we obtain 

^<l^+2F=%ir, + ^,^, + 



dt 



(8). 



In this the first term represents the rate at which energy is 
being stored in the system ; 2-P'is the rate of dissipation ; and the 
two together account for the work done upon the system by the 
extemfitl forces.] 



83.] COEXISTENCE OF SMALL MOTIONS. 105 

83. Before proceeding further, we may draw an important 
inference from the linearity of our equations. If corresponding 
respectively to the two sets of forces ^j, ^j,..., ^Z, ^j', ... two 
motions denoted by -^i, "^j, ..., -^Z, -^a', ... be possible, then must 
also be possible the motion i^i + -^/r/, i^, + 1^/, ... in conjunction 

with the forces ^i + ^/, ^j + ^a' Or, as a particular case, 

when there are no impressed forces, the superposition of any two 
natural vibrations constitutes also a natural vibration. This is the 
celebrated principle of the Coexistence of Small Motions, first 
clearly enunciated by Daniel Bernoulli. It will be understood 
that its truth depends in general on the justice of the assumption 
that the motion is so small that its square may be neglected. 

[Again, if a system be under the influence of constant forces 
"^i, &c., which displace it into a new position of equilibrium, the 
vibrations which may occur about the new position are the same 
as those which might before have occurred about the old position.] 

84. To investigate the free vibrations, we must put ^i, "^j, ... 
equal to zero ; and we will commence with a system on which no 
frictional forces act, for which therefore the coefficients e^t , &c. are 
even functions of the symbol D, We have 

^1^1 + ^n'^s + • • • =0 \ 



From these equations, of which there are as many (m) as the 
system possesses degrees of liberty, let all but one of the variables 
be eliminated. The result, which is of the same form whichever 
be the co-ordinate retained, may be written 

V^ = ; (2), 

where V denotes the determinant 



(3), 



I 

^M> ^aa> ^2S» ••» 
^n» ^J2» ^w» ••• 



and is (if there be no friction) an even function of D of degree 2//i. 
Let ±Xi, IX,, ..., ±Xtn be the roots of V=0 considered as an 
equation in D, Then by the theory of differential equations the 
most general value of ^ is 

^ = il€^>^ + ^'e-*>< + 5€^^ + £'e-^+ (4), 



106 VIBRATING SYSTEMS IN GENERAL. [84. 

where the 2m quantities A, A\B, B\ &c. are arbitrary constants. 
This form holds good for each of the co-ordinates, but the constants 
in the different expressions are not independent. In fact if a 
particular solution be 

yJTi = ili€^»^ ylt2 = A2^'^, &c., 

the ratios Ai : A^ : A^... are completely determined by the 

equations 

e„ili + eij^j4-ei8il,4- =0 ^ 

6jnili + «2a^2 + ^»^s+ =0 > (5), 

where in each of the coefficients such sls Cn^Xi is substituted for D. 
Equations (5) are necessarily compatible, by the condition that Xi 
is a root of V = 0. The ratios A^ :A^ :A^ ,.. corresponding to 
the root —X, are the same as the ratios AixA^'^A^.,,, but for 
the other pairs of roots Xa, — Xa, &c. there are distinct systems of 
ratios. 

86. The nature of the system with which we are dealing 
imposes an important restriction on the possible values of X. If Xj 
were real, either Xi or — Xi would be real and positive, and we 
should obtain a particular solution for which the co-ordinates, and 
with them the kinetic energy denoted by 

Xi» [\ckiAi^ + . . . ai^A lilj + . . . } €*2^'<, 

increase without limit. Such a motion is obviously impossible for 
a conservative system, whose whole energy can never differ from 
the sum of the potential and kinetic energies with which it was 
animated at starting. This conclusion is not evaded by taking Xi 
negative ; because we are as much at liberty to trace the motion 
backwards as forwards. It is as certain that the motion never was 
infinite, as that it never will he. The same argument excludes the 
possibility of a complex value of X. 

We infer that all the values of X are purely imaginary, cor- 
responding to real negative values of X'. Analytically, the fact 
that the roots of V = 0, considered as an equation in i>, are 
all real and negative, must be a consequence of the relations 
subsisting between the coefficients On, Oij, ..., Cn, Cia, ... in virtue of 
the fact that for all real values of the variables T and V are 
positive. The case of two degrees of liberty will be afterwards 
worked out in full. 



86.] NORMAL CO-ORDINATES. 107 

86. The form of the solution may now be advantageously 

changed by writing irii for \i, &c. (where i=\/ — 1), and taking 
new arbitrary constants. Thus 

>^i = Ai cos {lilt — a) +A cos {nj; — /8) + Oi cos (nj; — 7) + ... ^ 

-^a = -4a COS (n^t — a) + ^2 cos {n^ — /8) + CjCos {nJ; — 7) -f ... ...,(1) 

-^3 = A^ cos (riit — a) + 5, cos (fi^t — ^)-¥Ci cos {nJL — 7) -f . . . 

where n^, n^, &c. are the m roots of the equation of m*^ degree 
in n' found by writing — n* for jD' in V = 0. For each value of n 
the ratios Ai-.A^iA^,., are determinate and real. 

This is the complete solution of the problem of the free 
vibrations of a conservative system. We see that the whole 
motion may be resolved into m normal harmonic vibrations of 
(in general) different periods, each of which is entirely indepen- 
dent of the others. If the motion, depending on the original 
disturbance, be such as to reduce itself to one of these (rij) 
we have 

•^/tj = J. J cos {rtit — a), -^/ra = A^ cos {n^t - a), &c (2), 

where the ratios A^: A^: A^ ,,. depend on the constitution of the 
system, and only the absolute amplitude and phase are arbitrary. 
The several co-ordinates are always in similar (or opposite) phases 
of vibration, and the whole system is to be found in the configura- 
tion of equilibrium at the same moment. 

We perceive here the mechanical foundation of the supremacy 
of harmonic vibrations. If the motion be sufficiently smcdl, the 
differential equations become linear with constant coefficients ; 
while circular (and exponential) functions are the only ones which 
retain their type on differentiation. 

87. The m periods of vibration, determined by the equation 
V =0, are quantities intrinsic to the system, and must come out 
the same whatever co-ordinates may be chosen to define the con- 
figuration. But there is one system of co-ordinates, which is 
especially suitable, that namely in which the normal t)rpes of 
vibration are defined by the vanishing of all the co-ordinates but 
one. In the first type the original co-ordinates '^ij'^a* &c. have 
given ratios; let the quantity fixing the absolute values be ^1, so 
that in this type each co-ordinate is a known multiple of ^,. So 
in the second type each co-ordinate may be regarded as a known 
multiple of a second quantity ^21 &nd so on. By a suitable deter- 



(1), 



108 VIBRATING SYSTEMS IN GENERAL. [87. 

minatioQ of the m quantities <^, ^2, &c., any configuration of the 
system may be represented as compounded of the m configurations 
of these types, and thus the quantities ^ themselves may be looked 
upon as co-ordinates defining the configuration of the system. 
They are called the normal co-ordinates \ 

When expressed in terms of the normal co-ordinates, T and V 
are reduced to sums of squares ; for it is easily seen that if the 
products also appeared, the resulting equations of vibration would 
not be satisfied by putting any m — 1 of the co-ordinates equal to 
zero, while the remaining one was finite. 

We might have commenced with this transformation, assuming 
from Algebra that any two homogeneous quadratic functions can 
be reduced by linear transformations to sums of squares. Thus 

where the coefficients (in which the double suffixes are no longer 
required) are necessarily positive. 
Lagrange's equations now become 

ai^+Ci^i = 0, Oa^a + Ca<^ = 0, &C (2), 

of which the solution is 

4>i = A cos(ni^ — a), ^ = 5 cos (^i^^ — /8), &c (3), 

where A^ £..., a, 13... are arbitrary constants, and 

7ii'=Ci-^ai, Wa' = C2-r-aj, &c (4). 

[The vibrations expressed by the various normal co-ordinates 
are completely independent of one another, and the energy of the 
whole motion is the simple sum of the parts corresponding to the 
several normal vibrations taken sepcirately. In fact by (1) 

r+F=ic,il,« + ic^^« + (5). 

By the nature of the case the coefficients a are necessarily 
positive. But if the equilibrium be unstable, some of the 
coefficients c may be negative. Corresponding to any negative 
c, n becomes imaginary and the circular functions of the time are 
replaced by exponentials. 

In any motion proportional to e^ the disturbance is equally 
multiplied in equal times, and the degree of instability may be 
considered to be measured by \. If there be more than one 

1 Thomson and Tait*B Natural Philosophy, first edition 1867. § 337. 



87.] PERIODS OF FREE VIBRATIONS. 109 

unstable mode, the relative importance is largely determined by 
the corresponding values of X. Thus, if 

in which Xi > Xa» then whatever may be the finite ratio of A : B, 
the first term ultimately acquires the preponderance, inasmuch as 

Ae^^^ :B^^^{AIB)e^'^^-^^K 

In general, unstable equilibrium when disturbed infinitesimally 
will be departed from according to that mode which is most 
unstable, viz. for which \ is greatest. In a later chapter we sh^U 
meet with interesting applications of this principle. 

The reduction to normal co-ordinates allows us readily to trace 
what occurs when two of the values of n' become equal. It is 
evident that there is no change of form. The spherical pendulum 
may be referred to as a simple example of equal roots. It is 
remarkable that both Lagrange and Laplace fell into the error of 
supposing that equality among roots necessarily implies terms 
containing ^ as a factor^ The analytical theory of the general 
case (where the co-ordinates are not normal) has been discussed by 
Somof* and by Routh'.] 

88. The interpretation of the equations T>f motion leads to a 

theorem of considerable importance, which may be thus stated*. 

The period of a conservative system vibrating in a constrained type 

about a position of stable equilibrium is stationary' in value when 

the type is normal. We might prove this from the original 

equations of vibi-ation, but it will be more convenient to employ 

the normal co-ordinates. The constraint, which may be supposed 

to be of such a character as to leave only one degree of freedom, is 

represented by taking the quantities ^ in given ratios. 

If we put 

4>, = A,e, 4>^^A,e,kc (1), 

^ is a variable quantity, and Ai, A^, &c. are given for a given con- 
straint. 

The expressions for T and V become 

r={iaiili» + iM2' + }^^ 

V ^ [^c,A,' + ^cj.^ -\' }^, 

1 Thomson and Tait, 2nd edition, § 343 m. 

2 St Petenb, Acad. Scu Mhn, i. 1859. 

> Stability of Motion (Adams Prize Essay for 1877). See also,Routh*s Rigid 
Dynamict, 5th edition, 1892. 

* Proeeedingt of the Mathematical Society, June, 1873. 



110 VIBRATING SYSTEMS IN GENERAL. [88. 

whence, if varies as cos pt, 

P " a^Ai'-j-a^Aj' + :..'{' OmAn,' ^ ^' 

This gives the period of the vibration of the constrained type ; 
and it is evident that the period is stationary, whefn all but one of 
the coefficients Ai.A^,... vanish, that is to say, when the type 
coincides with one of those natural to the system, and no constraint 
is needed. 

[In the foregoing statement the equilibrium is supposed to be 
thoroughly stable, so that all the quantities c are positive. But 
the theorem applies equally even though any or all of the c*s be 
negative. Only if p^ itself be negative, the period becomes 
imaginary. In this case the stationary character attaches to the 
coefficients of t in the exponential terms, quantities which measure 
the degree of instability. 

Corresponding theorems, of importance in other branches of 
science, may be stated for systems such that only T and F, or only 
Fand F, are sensible \ 

The stationary property of the roots of Lagrange's determinant 
(3) § 84, suggests a general method of approximating to their 
values. Beginning with assumed rough approximations to the 

ratios AiiA^iA^ we may calculate a first approximation to 

jp^from 

■^ Clii-A 1 "^ 8 ^^^a3-"-a "H • • • "H UiiAiJL J 4" . . . 

With this value of p* we may recalculate the ratios AiiA^... from 
any (m — 1) of equations (5) § 84, then again by application of (3) 
determine an improved value of jp*, and so on.] 

By means of the same theorem we may prove that an increase 
in the mass of any part of a vibrating system is attended by a 
prolongation of all the natural periods, or at any rate that no 
period can be diminished. Suppose the increment of mass to be 
infinitesimal. After the alteration, the types of fi:^e vibration will 
in general be changed ; but, by a suitable constraint, the system 
may be made to retain any one of the former types. If this be 
done, it is certain that any vibration which involves a motion of 
the part whose mass has been increased will have its period 
prolonged. Only as a particular case (as, for example, when a 
load is placed at the node of a vibrating string) can the period 

1 Brit, A$$, Rep. for ISSo, p. 911. 



88.] PERIODS OF FREE VIBRATIONS. HI 

remain unchanged. The theorem now allows us to assert that 
the removal of the constraint, and the consequent change of type, 
can only affect the period by a quantity of the second oi*der ; and 
that therefore in the limit the free period cannot be less than 
before the change. By integration we infer that a finite increase 
of mass must prolong the period of every vibration which involves 
a motion of the part affected, and that in no case can the period 
be diminished ; but in order to see the correspondence of the two 
sets of periods, it may be necessary to suppose the alterations 
made by steps. Conversely, the effect of a removal of part of 
the mass of a vibrating system must be to shorten the periods 
of all the free vibrations. 

In like manner we may prove that if the system undergo such 
a change that the potential energy of a given configuration is 
diminished, while the kinetic energy of a given motion is unaltered, 
the periods of the free vibrations are all increased, and conversely. 
This proposition may sometimes be used for tracing the effects of 
a constraint; for if we suppose that the potential energy of 
any configuration violating the condition of constraint gradually 
increases, we shall approach a state of things in which the 
condition is observed with any desired degree of completeness. 
During each step of the process every free vibration becomes ] 
(in general) more rapid, and a number of the free periods (equal 
to the degrees of liberty lost) become infinitely small. The 
same practical result may be reached without altering the po- 
tential energy by supposing the kinetic energy of any moticm 
violating the condition to increase without limit. In this case 
one or more periods become infinitely large, but the finite 
periods are ultimately the same as those arrived at when the 
potential energy is increased, although in one case the periods 
have been throughout increasing, and in the other diminishing. 
This example shews the necessity of making the alterations by 
steps; otherwise we should not understand the correspondence 
of the two sets of periods. Further illustrations will be given 
under the head of two degrees of freedom. 

By means of the principle that the value of the free periods 
is stationary, we may easily calculate corrections due to any 
deviation in the system from theoretical simplicity. If we take 
as a hypothetical type of vibration that proper to the simple 
system, the period so found will differ from the truth by quan- 
tities depending on the squares of the irregularities. Several 



112 VIBRATING SYSTEMS IN GENERAL. [88. 

examples of such calculations will be given in the course of 
this work. 

89. Another point of importance relating to the period of a 
system vibrating in an arbitrary type remains to be noticed 
It appears from (2) § 88, that the period of the vibration cor- 
responding to any hypothetical type is included between the 
greatest and least of those natural to the system. In the case 
of systems like strings and plates which are treated as capable 
of continuous deformation, there is no least natural period; 
but we may still assert that the period calculated from any hypo- 
thetical type cannot exceed that belonging to the gravest normal 
tjrpe. When therefore the object is to estimate the longest 
proper period of a system by means of calculations founded 
on an assumed type, we know a priori that the result will come 
out too small. 

In the choice of a hypothetical type judgment must be 
used, the object being to approach the truth as nearly as can 
be done without too great a sacrifice of simplicity. Thus the 
type for a string heavily weighted at one point might suitably 
be taken from the extreme case of an infinite load, when the 
two parts of the string would be straight. As an example of 
a calculation of this kind, of which the result is known, we 
will take the case of a uniform string of length I, stretched 
with tension Ti, and inquire what the period would be on 
certain suppositions as to the type of vibration. 

Taking the origin of x at the middle of the string, let the 
curve of vibration on the positive side be 



y = co8/)f|l-(^)"| (1), 

and on the negative side the image of this in the axis of y, 
n being not less than unity. This form satisfies the condition 
that y vanishes when a? = ± ^i. We have now to form the ex- 
pressions for T and V, and it will be sufficient to consider the 
positive half of the string only. Thus, p being the longitudinal 
density, 



89.] PERIODS OF FREE VIBRATIONS. 113 

He^ce ^.'J."^^>.| (., 

I{ n=l, the string vibrates as if the mass were concentrated 
in its middle point, and 

If n = 2, the form is parabolic, and 

^= pir- 

The true value of p* for the gravest t3rpe is — j- , so that 

the assumption of a parabolic form gives a period which is too 
small in the ratio tt : VlO or '9936 : 1. The minimum of p^, 
as given by (2), occurs when n = J (\/6 + 1) = 1*72474, and gives 

©« = 9-8990 5- 

The period is now too small in the ratio 

TT : \^9W90 = -99851 : 1. 

It will be seen that there is considerable latitude in the 
choice of a type, even the violent supposition that the string 
vibrates as two straight pieces giving a period less than ten 
per cent, in error. And whatever type we choose to take, the 
period calculated from it cannot be greater than the truth. 

[In the above applications it is assumed that there are no 
unstable modes. When unstable modes exist, the statement is 
that a constrained mode if stable possesses a frequency of vibra- 
tion less than that of the highest normal mode, and if unstable 
has a degree of instability less than that of the most unstable 
normal mode.] 

90. The rigorous determination of the periods and types of 
vibration of a given system is usually a matter of great difficulty, 
arising from, the fact that the functions necessary to express the 
modes of vibration of most continuous bodies are not as yet recog- 
nised in analysis. It is therefore o£ben necessary to fall back on 
methods of approximation, referring the proposed system to some 
other of a character more amenable to analysis, and calculating 
corrections depending on the supposition that the differeiice be- 
tween the two systems is small. The problem of approximately 

R. 8 



114 VIBRATING SYSTEMS IN GENERAL. [90, 

simple systems is thus one of great importance, more especially 
as it is impossible in pi-actice actually to realise the simple forms 
ftl)OUt which we can most easily reason. 

Let us suppose then that the vibrations of a simple system are 
thoroughly known, and that it is required to investigate those 
of a system derived from it by introducing small variations in 
the mechanical functions. If 0i, 0,, &c. be the normal co-ordi- 
nates of the original system, 

and for the varied system, referred to the same co-ordinates, 
which are now only approximately normal, 

F+Sr=i (Cx + SCn) </>i»-h ... + SCi,</h<N 
in which Son, Set,,, Scu, Scj,, &c. are to be regarded as small 
quantitiea In certain cases new co-ordinates may appear, but 
if so their coeflScients must be small. From (1) we obtain for the 
Lagrangian equations of motion, 

(oi + Sou J5* + Ci + Scu) <^ + (Soi^Ifi -f Scij) <f>2 ^ 

+ (Sa,82)«4-Sc,s)0,4- ... =0 

(2). 



■""1 (1), 

"i" • • . / 



-h(Safl2)^ + &a)<^8 + ... =0 

In the original system the fundamental types of vibration 
are those which correspond to the variation of but a single co- 
ordinate at a time. Let us fix our attention on one of them, 
involving say a variation of 0^, while all the remaining co- 
ordinates vanish. The change in the system will in general 
entail an alteration in the fundamental or normal types; but 
under the circumstances contemplated the alteration is small. 
The new normal type is expressed by the sjmchronous variation 
of the other co-ordinates in addition to 0r ; but the ratio of any 
other <f>, to <f>r is small. When these ratios are known, the normal 
mode of the altered system will be determined. 

Since the whole motion is simple harmonic, we may suppose 
that each co-ordinate varies as eosj^r^, and substitute in the 
differential equations —pr^ for L^. In the s^^ equation <f>g occurs 
with the finite coefficient 

— ttfPr^ — ha^Pr + c, + 8c«. 



90.] APPROXIMATELY SIMPLE SYSTEMS. 115 

The coefficient of ^^ is 

The other terms are to be neglected in a first approximation, 
since both the co-ordinate (relatively to 0^) &nd its coefficient are 
small quantities Hence 

<^.:<^, = -.«^---:Prg^ (3). 

Now — a,p,* + c, = 0, 

-"""fc" i"--*"^^ <*)• 

the required result. 

If the kinetic energy alone undergo variation, 

^,:^^=_P!!_«^ (5). 

The corrected value of the period is determined by the rth 
equation of (2), not hitherto used. We may write it, 

Substituting for ^, : ^^ from (4), we get 

^•' Or + fiOrr dfi^ip^-p*) 

The first term gives the value of pr^ calculated without allow- 
ance for the change of type, and is sufficient, as we have already 
proved, when the square of the alteration in the system may 
be neglected. The terms included under the symbol % in 
which the summation extends to all values of a other than r, 
give the correction due to the change of type and are of the 
second order. Since a« and Or are positive, the sign of any term 
depends upon that o{ pi^—pr*. li p^>pr*, that is, if the mode 
s be more acute than the mode r, the correction is negative, 
and makes the calculated note graver than before; but if the 
mode 8 be the graver, the correction raises the note. If r refer 
to the gravest mode of the system, the whole correction is 
negative ; and if r refer to the acutest mode, the whole correction 
is positive, as we have already seen by another method. 

91. As an example of the use of these formulae, we may 
take the case of a stretched string, whose longitudinal density p 
is not quite constant. If x be measured from one end, and y 



116 VIBRATING SYSTEMS IN GENERAL. [91. 

be the transverse displacement, the configuration at any time t 

will be expressed by 

. . TTX , 27rx , . Ztrx ,, . 

y = </>i8in ^ -h<^sin— ^ -h<^8 8in-|- + (1), 

I being the length of the string. <f>^, 02,... are the normal 
co-ordinates for /» = constant, and though here p is not strictly 
constant, the configuration of the system may still be expressed 
by means of the same quantities. Since the potential energy 
of any configuration is the same as if p = constant, SF= 0. For 
the kinetic energy we have 

T + hT = \\ pfi^ism -^-h^asm -, +...jda? 

t ; • r^ • o'wa? , 1 ; • /' • ^2irx , 
= J</>i I psin'-^ aa7 + i4>a' I psm'-^- cw?4-... 

y ^ .'0 ^ 

; : [^ . TTX . 27rX , 

4-9i9a I psm-y-sm — y cto4- .... 

If p were constant, the products of the velocities would 
disappear, since ^, 0„ &c. are, on that supposition, the normal 
co-ordinates. As it is, the integral coefficients, though not actually 
evanescent, are small quantities. Let p = Po + ^P > then in our 
previous notation 

Or = i lpo> ^Orr = / Sp siu^ — y- dx, SOrg = I Sp siu , - siu - .- dx, 

Jo.*' Jo ^ ^ 

Thus the type of vibration is expressed by 

or, since pr^ : pt^ = r^ : $-, 

r* f'2Sp . rvx . sttx , ,.. 

0. : '^' = ^^j;/^ «>«»-; «»°— rf^ (2). 

Let us apply this result to calculate the displacement of the 
nodal point of the second mode (r = 2), which would be in the 
middle, if the string were uniform. In the neighbourhood of 
this point, if x = ^l + Sx, the approximate value of y is 

. TT , . . 27r . Stt 

y = <^ism2 +<^sm Y + <^,8m g +••• 



^ (tt . TT 27r 2Tr , 1 

ox <j <t>i COH - -j- y 92C0S-^ -h ... h 



TT 



= 01-08 + 05- ••• +7 Sir {-205 + 204+ ...}. 



91.] EXAMPLES. 117 

Hence when y = 0, 

&^=2;^J*^"*»'^*»"-J ^^^ 

approximately, where 

To shew the application of these formulae, we may suppose 
the irregularity to consist in a small load of mass po\ situated 
at a? = ii, though the result might be obtained much more easily 
directly. We have 

. _ 2\ f 2^ __2 2 2_ ) 

7rV2tP-4 3^-4 58-4"*" 7»-4'*" j' 

from which the value of Sx may be calculated by approximation. 
The real value of Bx is, however, very simple. The series within 
brackets may be written 

-f +3-5-7 + 9 + n-*"j' 
which is equal to 

ol + a?* 
The value of the definite integral is 

TT -T- 4 sm T , 
4 

and thus oa?= -. . ^- = — 75, 

7rv2 4 2 

as may also be readily proved by equating the periods of vibra- 
tion of the two parts of the string, that of the loaded part being 
calculated approximately on the assumption of unchanged type. 

As an example of the formula (6) § 90 for the period, we 
may take the case of a string carrying a small load p^ at its 
middle point. We have 

ar = ^lpo» Sorr = /^o^ sin' -g- , Sor, = p©^ siu -g- sin -g- , 

and thus, if P^ be the value corresponding to \ = 0, we get when 
r is even, pr = Pr, and when r is odd, 



-/ 



'-•-'■'•{uwr'^i^v} «■ 



1 Todhunter's Int. Cale, % 255. 



118 VIBRATING SYSTEMS IN GENERAL. [91. 

where the summation is to be extended to all the odd values 
of 8 other than r. If r = 1, 



p,« = P,«|l- 



2X 4\» ^ 4 VI 



Now 22-T^ = 2-.-2 ^ 



««-l «-l « + l' 
in which the values of « are 3, 5, 7, 9.... Accordingly 

«»-l 4' 
and p.. = p,.|i_^ + ^V I (6), 

giving the pitch of the gravest tone accurately as far as the 
square of the ratio X : L 

In the general case the value of pr^, correct as far as the first 
order in Sp, will be 

^.=P,{i-^}.i.,|i-l0^.-4...(,, 

92. The theory of vibrations throws great light on expansions 
of arbitrary functions in series of other fiinctioDS of specified 
types. The best known example of such expansions is that 
generally called after Fourier, in which an arbitrary periodic 
function is resolved into a series of harmonics, whose periods 
are submultiples of that of the given function. It is well known 
that the diflSculty of the question is confined to the proof of the 
possibility of the expansion ; if this be assumed, the determination 
of the coeflScients is easy enough. What I wish now to draw 
attention to is, that in this, and an immense variety of similar 
cases, the possibility of the expansion may be inferred fi:*om 
physical considerations. 

To fix our ideas, let us consider the small vibrations of a 
imiform string stretched between fixed points. We know from 
the general theory that the whole motion, whatever it may 
be, can be analysed into a series of component motions, each 
represented by a harmonic function of the time, and capable 
of existing by itself. If we can discover these normal types, 
we shall be in a position to represent the most general vibration 
possible by combining them, assigning to each an arbitrary 
amplitude and phase. 



92.] NORMAL FUNCTIONS. 119 

Assuming that a motion is harmonic with respect to time, 
we get to determine the type an equation of the form 

whence it appears that the normal functions are 

. irx . ^irx . Sirx « 

y = 8m-y, y = 8m—j-, y = sm-j-,&c. 

We infer that the most general position which the string can 
assume is capable of representation by a series of the form 



J . wa? . , . 2'rrx . . ^irx 
Ai sm -y- + ila sm ~^— + iS, sm -v— 



which is a particular case of Fourier's theorem. There would 
be no difficulty in proving the theorem in its most general form. 

So feu: the string has been supposed uniform. But we have 
only to introduce a variable density, or even a single load at 
any point of the string, in order to alter completely the ex- 
pansion whose possibility may be inferred from the dynamical 
theory. It is unnecessary to dwell here on this subject, as 
we shall have further examples in the chapters on the vibrations 
of particular systems, such as bars, membranes, and confined 
masses of air. 

92 a. In § 88 we have a formula for the frequency of vibration 
applicable when by the imposition of given constraints the original 
system is left with only one degree of freedom. It is of interest 
to trace also the effect of less complete constraints, such as may 
be expressed by linear relations among the normal co-ordinates of 
number less by at least two than that of the (original) degrees of 
freedom. Thus we may suppose that 

S^i<Ai + S^t.<^+5's<A. + ... = I (1). 

hi<^ + Aj<^ -h Aa^s -f- . . . = 



If the number of equations (r) fall short of the number of the 
degrees of freedom by unity, the ratios 0i:^:09... are fully 
determined, and the case is that of but one outstanding degree of 
freedom discussed in § 88. 

This problem may be treated in more than one way, but the 



120 VIBRATING SYSTEMS IN GENERAL. [92 a. 

most instructive procedure is to trace the effect of additions to T 
and V. We will suppose that equations (1) § 87 are altered to 

r=K<^i' + W»" + ---+i«(/i<^i •+•/«*« + •••y (2), 

and that -P, not previously existent, is now 

J^=i)8(/,^i+/s<^,+ . ..)».:...; (4). 

The connection with the proposed problem will be understood 
by supposing for instance that a = 0, y8 = 0, while 7 = x . By (3) 
the potential energy of any displacement violating the condition 

/i*i+/20,+ ...=O (5) 

is then infinite, and this is tantamount to the imposition of the 
constraint represented by (5). 

Lagrange's equations with \ written for D now become 



....(6). 



If we multiply the first of these by/i/(aiV4- Cj), the second by 
fij(<h^^ + Cj)* ^^^ s^ ^^» ^^^ ^d the results together, the factor 
(/i^+/202 + *'* ) ^^1 divide out, and the determinant takes the 
form 

^'^ + .# + + ..vAv-rT = (7). 



If any one of the quantities a, )8, 7 become infinite while the 
others remain finite, the effect is equivalent to the imposition of the 
constraint (5), and the result may be written 

2/V(a>-* + c) = (8V. 

When multiplied out this equation is of degree (m — 1) in V, one 
degree of freedom having been lost. 

If we put )8 = 0, (7) is an equation of the mth degree in V, and 
the coefiBcients a, 7 enter in the same way as do Oi, Ci; a,, Cj; &c. 

In order to refer more directly to the case of vibrations about 
stable equilibrium, we will write p* for — V. The values of pr 
belonging to the unaltered system, viz. m,', n^^..., are given as 
before by 

Ci — aiWi' = 0, Cj — 02^2^ = 0, &c., (9); 

and we will also write 

y-ap'^O (10), 

1 Booth's Rigid Dynamics, 5th edition, 1892, § 67. 



92 a.] ONE CONSTRAINT. 121 

where i/* relates to the supposed additions to T and V considered 
as belonging to an independent vibrator. Let the order of magni- 
tude of these quantities be 

Wi', ?laS V, »^, fh+i W,n* (11). 

We shall see that there is a root of (7) between each consecutive 
pair of the quantities (11). 

Our equation may be written 

/i'(7-ap»)(c-a,p«)(c-a,p«) 

+/i'(7-ap')(ci~a^)(c,-a,p») 

+ 

-f (Ci-a,p«)(c-a,p«) = (12). 

When p" coincides with any of the quantities (11), all but one 
of the terms in (12) vanish, and the sign of the expression is the 
same as that of the term which remains over. When p' < Wi', all 
the terms are positive, so that there is no root less than n^. 
When ^ = n,^ the expression (12) reduces to the positive quantity 

/i* (7 - awi*) (Cs - «iWi') {Cz - asw,«) 

When p^ rises to n^, (12) becomes 

/a' (7 - aO (Ci - Oi^aO (Cj - Oj^ij') ; 

and this is negative, since the factor (c^ — (h^) is now negative. 
Hence there is a root of (12) between n,' and nf. When J3'= w,^, 
the expression is again positive, and thus there is a root between 
n,* and n^. This argument may be continued, and it proves that 
there is a root of (12) between any consecutive two of the (m-|- 1) 
quantities (11). The m roots of (12) are now accounted for, and 
there is none greater than n„^. If we compare the values of the 
roots before and after the change, we see that the eflFect is to 
cause a movement which is in every case towards i^.^ Considered 
absolutely the movement is in one direction for those roots that 
are greater than ir and in the opposite direction for those that 
are less than i;*. Accordingly the interval from rir* to rir+i', in 
which v^ lies, contains after the change two roots, one on either 
side of v'. 

If I/* be less than any of the quantities n', as happens when 
7 = 0, one root lies between p* and Wi*, one between Wi'^ and n,*, and 
so on. Thus every root is depressed. On the other hand if 
^ > ^^ every root is increased. This happens if a = 0. (§ 88.) 

^ It will be understood that in particular cases the movement may vanish. 



122 VIBRATING SYSTEMS IN GENERAL. [92 a. 

The results now arrived at are of course independent of the 
special machinery of normal co-ordinates used in the investigation. 

If to any part of a system {ni\ n^ ) be attached a vibrator 

(y) having a aingle degree of freedom, the eflFect is to displace all 
the quantities Wj*, ... in the direction of i/". Let us now suppose 
that a second change is made in the vibrator whereby a becomes 
a 4- a', and 7 becomes 7 + 7'. Every root of the determinantal 
equation moves towards j/'', where 7' — aV' = 0. If we suppose 
that V* = 1/*, the movements are in all cases in the same directions 
as before. Going back now to the original system, and supposing 
that a, 7 grow from zero to their actual values in such a manner 
that i^ remains constant, we see that during this process the roots 
move without regression in the direction of closer agreement 
with v*. 

As a and 7 become infinite, one root of (12) moves to coinci- 
dence with 1/", while the remaining (m — 1) roots, corresponding to 
the constrained system, are given by 

2/V(c-ap') = (13), 

and are independent of the value of i/*. 

Particular cases are obtained by supposing either i/* = 0, or 
!/> = X . Whether the constraint is effected by making infinite 
the kinetic energy of any motion, or the potential energy of 
any displacement, which violates it, makes no difference to the 
vibrations which remain. In the first case one vibration becomes 
infinitely slow, and in the second case one becomes infinitely quick. 
However the constraint be arrived at, the (m — 1) frequencies of 
vibration of the constrained system separate^ the m frequencies 
of the original system. 

Any number of examples of this theorem may be invented 
without diflSculty. Consider the case of a uniform stretched 
string, held at both ends and vibrating transversely. This is the 
original system. Now introduce a constraint by holding at rest a 
point which divides the length in the proportion (say) of 3 : 2. 
The two parts vibrate independently, and the frequencies for each 
part form an arithmetical progression. If the frequencies proper 
to the undivided string be 1, 2, 3, 4 ; those for the parts are 

^ But in particular oases the " separation *' may vanish. The theorem in the 
text was proved for two degrees of freedom in the first edition of this work. In 
its generality it appears to be due to Bouth. 



92 a.] ONE CONSTRAINT. 123 

f (1, 2, 3, ...) and f (1, 2, 3, ...). The beginning of each series is 
shewn in the accompanying scheme ; 

123456789 10 11 12 






H "" 10 



111 



I IJ 3J »■ 5 63 

I 2J 5 -^ 7i 10 -^ 

and it will be seen that between any consecutive numbers in 
the first row there is a number to be found either in the second 
or in the third row. In the case of 5 and 10 we have an extreme 
condition of things ; but the slightest displacement of the point 
at which the constraint is applied will displace one of the fives, 
tens &c. to the lefb and the other to the right. 

The coincidences may be avoided by dividing the string 
incommensurably. Thus» if x be an incommensurable number 
less than unity, one of the series of quantities m/x, m/(l —a?), where 
m is a whole number, can be found which shall lie between any 
given consecutive integers, and but one such quantity can be found. 

Again, let us suppose that a system is referred to co-ordinates 
which are not normal (§ 84), and let the constraint represented by 
-^1 = be imposed. The determinant of the altered system is 
formed from that of the original system by erasing the first row 
and the first column. It may be called V^, and from this again 
may be formed in like manner a new determinant Vj, and so on. 
These determinants form a series of functions of p^, regularly 
decreasing in degree; and we conclude that the roots of each 
separate the roots of that immediately preceding ^ 

It may be remarked that while for the sake of simplicity of 
statement we have supposed that the equilibrium of the original 
system was thoroughly stable, as also that of the vibration brought 
into connection therewith, these restrictions may easily be 
dispensed with. In any case the series of positive and negative 

quantities, ni*, n,*, and i/*, may be arranged in algebraic order, 

and the effect of the vibrator is to cause a movement of every 
value of/)* in the direction of i/*. 

In order to extend the above theory we will now suppose that 
the addition to T is 

i^/(fi4>i +/2^a +•••)' + H (9i4>i + .92</>a +...)' 

■^ioLH{lh4>^ + h4,+ ...y+ (14) 

1 Eouth*s Rigid Dynamics, 5th edition. Part ii. § 5S. 



124 



VIBRATING SYSTEMS IN GENERAL. 



[92 a. 



and the addition to V 

I£ W6 set 

a/K^ + rif^F'. a,X^ + yg=G', (16), 

and so on, Lagrange's equations become 

(a,X' + c,) ^ + Fy\ (/,^ +/j<^ + . . .) 
+ O'gi(gi<l>i+9i<f>i+"-) + S\0h^ + h^ + ...) + ... = 0...(17), 

(o^* + c,) ^ + F'f, {fi<f>, +f^^ + ...) 

+ 0'gt{gi^+g^+ •••) + H'ht{K4>i + h^t + ...) + — = o...(i8), 

and so on, the number of equations being equal to the number 
(m) of co-ordinates ^, ^, .... The number of additions (r), corre- 
sponding to the letters y, g,h,...,\s supposed to be less than m. 

From the above m equations let r new ones be formed, as 
follows. For the first multiply (17) by //(oiX" + c,), (18) by 
/^/(OjX' + c,), and so on, and add the results together. For the 
second proceed in the same manner, using the multipliers 
^i/(aiX' + c,), gt/ic^^^ + c,), &C. In like manner for the third 
equation use h instead of g, and so on. In this way we obtain r 
equations which may be written 

P' (Ml +/A + • • •) IV^' + ^i" + P^ + -f** +. . -1 

+ G' igiih +g,4>t+-) {^i<?i + ^A + ■-} 

+ H' (h,<tf, + lii^ + ...){F,Hi + F^^+ ...} + =0...(19), 

+ 0'ig,<lH+g^ + ...) {1IG' + Gr'+ G,'+ ...) 

+ ^'(A,0, + A,(^,-l- ...) {G^H, + G^H, + ...) + =0...(20), 

and so on, where for brevity 

Ci' = <7iV(«i^* + <h), G^ = (///(a,X' + c), &c. I (21 ). 

^iG,=/iy./(o,V + c), &c. ) 

The determinantal equation, of the rth order, is thus 
1/J" + 2F», 1FG, tFH,... 

^FG, 1/G' + 10\ IGH, 

IFH, 1GH,1/H' + 1H*,... 



= 0, 



.(22). 



92 a.] 



SEVERAL CONSTRAINTS. 



125 



If, for example, there be two additions to T and V of the kind 
prescribed, the equation is 

p^ + ^, +-^ +2i».SG»-{2/'G}» = (23). 

and herein 

^Tt{F,Q,^F,G,y ^2-*). 

Equation (23) is in general of the mth degree in X', and 
determines the frequencies of vibration. In the extreme case 
where F' and 0' are made infinite, the system is subject to the 
two constraints 

S^i<^i + 5'i<^2 + ...=0J 
and the equation ^ giving the (771 — 2) outstanding roots is 



.(25), 



+ 



= 



.(26). 



(OjV + Ci) (a,X« 4- Cj) (aiX^* + Ci) (a,X» + c,) 

In general if the system be subject to the r constraints (1), the 
determinantal equation is 

'S.FF, IFO, IFH,... 
IFH, XGH, IHH,.., 



(27). 



If r be leas than m, this determinant can be resolved'* into a 
sum of squares of determinants of the same order (r). Thus if there 
be three constraints, the first of these squares is 

i\ jPa /; » 

G, G, (?, (28), 

Hi if a H^ 

and the others are to be found by including every combination of 
the m sufiBxes taken three together. To fall back upon the original 
notation we have merely in (28) to replace the capital letters 
F, (?,... by/, g,..., and to introduce the denominator 

(oiX* + c) (OaV + Cj) (a,X« + c). 

The determinantal equation for a system originally of m degrees 
of freedom and subjected to r constraints is thus found. Its form 



1 This resalt is due to Boatb, loc. cit. § 67. 
' Salmon, Leiso/ns on Higher Algebra^ § 24. 



126 



VIBRATING SYSTEMS IN GENERAL. 



[92 a. 



is largely determined by the consideration that it must remain un- 
aflfected by interchanges either of the letters or of the suffixes. 
That it would become nugatory if two of the conditions of con- 
straint coincided, could also have been foreseen. If r=m — 1, 
the system is reduced to one degree of freedom, and the equation 



IS 



f^ ft ji-" 
h^ As A4... 



(chy + Ci) + 



9i 9i 94"- 



(a,V + c) + ...=0 (29), 



in agreement with § (88). 

There are theories, parallel to the foregoing, for systems in 
which T and F, or V and F, are alone sensible. In these cases, if 
the functions be intrinsically positive, the normal motions are 
proportional to exponential functions of the time such as 6~*''. 
The quantities r^, r,,... are called the time-constants, or persis- 
tences, of the motions, being the times occupied by the motions in 
subsiding in the ratio of 6 : 1. The new persistences, after the 
introduction of a constraint, will separate the original values. 

The best illustrations of this theory are electrical, where the 
motions are not restricted to be small. Suppose (to take an 
electro-magnetic example) that in one branch of a net-work of 
conductors there is introduced a coil of persistence (when closed 
upon itself) equal to t', the original persistences being Ti, t,,.... 
Then the new persistences lie in all cases nearer to t\ and they 
separate the quantities t', Ti, tj.... If t' be made infinite as by 
increasing the self-induction of the additional coil without limit, 
or be made to vanish as by breaking the contact in the branch, 
the result is a constraint, and the new values of the persistences 
separate the former ones. 



93. The determination of the coefficients to suit arbitrarj' 
initial conditions may always be readily effected by the funda- 
mental property of the normal functions, and it maybe convenient 
to sketch the process here for systems like strings, bars, mem- 
branes, plates, &c. in which there is only one dependent variable 
(^ to be considered. If t^, t^...be the normal functions, and 
^, <^ . . . the corresponding co-ordinates. 



f=<^Wi + <^lt,+ 0,W,+ 



.(1). 



93.] INITIAL CX)NDITIONS. 127 

The equations of free motion are 

^ + ni><^ = 0, ^ + n,«<^ = 0,i&c (2), 

of which the solutions are 



(3). 



01 = ill sin riif 4- -Bi cos riit 
<l>i^ A^smnjt + B^ cos vjt 

The initial values of f and ^ are therefore 

t) = Wi-4iWi-fWjiljitj + Jvl8t£,+ ...J 

and the problem is to determine Ai, A^,,.. B^ B^.,. so as to 
correspond with arbitrary values of fo and ([q. 

li pdxhe the mass of the element da?, we have from (1) 

= i<f)i*L Mi'da: + J <^M/9 Ms'da? + ... + 01^ 1^ 

But the expression for T in terms of <^, <^3, &c. cannot contain 
the products of the normal generalized velocities, and therefore 
every integral of the form 

lptArt*/fo = (5). 



/' 



Hence to determine Br we have only to multiple the first 
of equations (4) by pibr and integrate over the system. We thus 
obtain 

BApUr^dx^ Ipur^ffib: (6). 

Similarly, rirAr \pUr^dx= IpUr^dx (7). 

The process is just the same whether the element dx he o, line, 
area, or volume. 

The conjugate property, expressed by (5), depends upon the 
fact that the functions u are normal. As soon as this is known 
by the solution of a differential equation or otherwise, we may 
infer the conjugate property without further proof, but the pro- 
perty itself is most intimately connected with the fundamental 
variational equation of motion § 94. 



128 VIBRATING SYSTEMS IN GENERAL. [94. 

94. If "T be the potential energy of deformation, f the 
displacement, and p the density of the (line, area, or volume) 
element dx, the equation of virtual velocities gives immediately 

BV+jp^S^dx=0 (1). 

In this equation 8F is a symmetrical function of f and Sf, 
as may be readily proved from the expression for V in terms 
of generalized co-ordinates. In fact if 

Suppose now that f refers to the motion corresponding to 
a normal function Ur, so that ^ + 7ir*f=0, while Sf is identified 
with another normal function Ug ; then 



S y = iir^ I ptirU/ix, 



Again, if we suppose, as we are equally entitled to do, that f 
varies as Ug and Sf as i/r, we get for the same quantity SF, 



r 

SV=ng^ jpUrU/lx; 



and therefore 



{nr^-ng^)jpurif4^ = (2), 



from which the conjugate property follows, if the motions re- 
presented respectively by Ur and u, have different periods. 

A good example of the connection of the two methods of 
treatment will be found in the chapter on the transverse vibrations 
of bars. 

96. Professor Stokes^ has drawn attention to a very general 
law connecting those parts of the free motion which depend 
on the initial displacements of a system not subject to frictional 
forces, with those which depend on the initial velocities. If 
a velocity of any type be communicated to a system at rest, 
and then after a small interval of time the opposite velocity 
be communicated, the effect in the limit will be to start the 
S3rstem without velocity, but with a displacement of the corre- 
sponding type. We may readily prove from this that in order 

1 Dynamical Theory of Diffraction, Cambridge Trans. Vol. ix. p. 1, 1866. 



95.] CX)NJUGATE PROPERTY. 129 

to deduce the motion depending on initial displacements from 
that depending on the initial velocities, it is only necessary to 
differentiate with respect to the time, and to replace the arbitrary 
constants (or functions) which express the initial velocities by 
those which express the corresponding initial displacement& 
Thus, if (f> be any normal co-ordinate satisfying the equation 

^ 4- n»<^ = 0, 

the solution in terms of the initial values of ^ and <^ is 

^ = ^cos7i^ + -<^osinn^ (1), 

of which the first term may be obtained from the second by 
Stokes' rule. 



R. ^ 



CHAPTER V. 

VIBBATING SYSTEMS IN GENERAL 

CONTINUED. 

96. When dissipative forces act upon a system, the character 
of the motion is in general more complicated. If two only of the 
functions T, F, and V be finite, we may by a suitable linear trans- 
formation rid ourselves of the products of the co-ordinates, and 
obtain the normal types of motion. In the preceding chapter we 
have considered the case of ^ = 0. The same theory with obvious 
modifications will apply when T = 0, or F=0, but these cases 
though of importance in other parts of Physics, such as Heat and 
Electricity, scarcely belong to our present subject. 

The presence of Motion will not interfere with the reduction of 
T and V to sums of squares ; but the transformation proper for 
them will not in general suit also the requirements of F, The 
general equation can then only be reduced to the form 

«i^i4-6u<^i4-6i2<^s + ... + Ci<^i = <E>i, &c (1), 

and not to the simpler form applicable to a system of one degree 
of freedom, viz. 

ai<if>i + 6i^ + Cj<^ = *i, &c. (2). 

We may, however, choose which pair of functions we shall 
reduce, though in Acoustics the choice would almost always fall 
on T and V. 

97. There is, however, a not unimportant class of cases in 
which the reduction of all three functions may be effected ; and 
the theory then assumes an exceptional simplicity. Under this head 
the most important are probably those when jP is of the same form 
as T or F. The first case occurs frequently, in books at any rate, 
when the motion of each part of the system is resisted by a re- 
tarding force, proportional both to the mass and velocity of the 



97.] T, F, V SIMULTANEOUSLY REDUCIBLE. 131 

part. The same exceptional reduction is possible when i^ is a 
linear function of T and F, or when T is itself of the same form as 
V. In any of these cases, the equations of motion are of the same 
form as for a system of one degree of freedom, and the theory 
possesses certain peculiarities which make it worthy of separate 
consideration. 

The equations of motion are obtained at once from T, F 
and V: — 

Oif 1 + bi4>i + Ci<l>, = *, , ) 

Oj^a-ffta^ + Cj^ = <E>a, &C. j ^ ^* 

in which the co-ordinates are separated. 

For the free vibrations we have only to put <E>i = 0, &c., and 
the solution is of the form 

(f> = e-i*< Uo ^^ + <l>o (coan't + ^, sin nftjl (2), 

where K=^b/a, n*=^c/a, n* = ^{n* — ^K*), 

and ^0 »nd <^o ^^re the initial values of <f> and ^. 

The whole motion may therefore be analysed into component 
motions, each of which corresponds to the variation of but one 
normal co-ordinate at a time. And the vibration in each of these 
modes is altogether similar to that of a system with only one 
degree of liberty. After a certain time, greater or less according 
to the amount of dissipation, the free vibrations become insignifi- 
cant, and the system returns sensibly to rest. 

[If F be of the same form as T, all the values of k are equal, 
viz. all vibrations die out at the same rate.] 

Simultaneously with the free vibrations, but in perfect inde- 
pendence of them, there may exist forced vibrations depending on 
the quantities 4>. Precisely as in the case of one degree of free- 
dom, the solution of 

a^4-6<^ + c^ = * (3) 

may be written 

<^ = i,f%-i*(/-r)sinn'(«-0*cft' (4), 

n J 

where as above 

K = b/a, n^ = c/a, vf = V(^' — i '<^)' 

To obtain the complete expression for ^ we must add to the 
right-hand member of (4), which makes the initial values of ^ 
and <l> vanish, the terms given in (2) which represent the residue 



132 VIBRATING SYSTEMS IN GENERAL. [97. 

at time t of the initial values <^o And ^o* If there be no firiction, 
the value of <f> in (4) reduces to 



4>^^l\mn(t-r)<Pdr (5). 



98. The complete independence of the normal co-ordinates 
leads to an interesting theorem concerning the relation of the 
subsequent motion to the initial disturbance. For if the forces 
which act upon the system be of such a character that they do no 
work on the displacement indicated by S^, then <l>i = 0. No such 
forces, however long continued, can produce any eflFect on the 
motion ^. If it exist, they cannot destroy it ; if it do not exist, 
they cannot generate it. The most important application of the 
theorem is when the forces applied to the system act at a node of 
the normal component <^, that is, at a point which the component 
vibration in question does not tend to set in motion. Two extreme 
cases of such forces may be specially noted, (1) when the force is 
an impulse, starting the system from rest, (2) when it has acted so 
long that the system is again at rest under its influence in a dis- 
turbed position. So soon as the force ceases, natural vibrations 
set in, and in the absence of friction would continue for an in- 
definite time. We infer that whatever in other respects their 
character may be, they contain no component of the type <^i. This 
conclusion is limited to cases where T, F, V admit of simultaneous 
reduction, including of course the case of no friction. 

99. The formulae quoted in § 97 are applicable to any kind of 
force, but it will often happen that we have to deal only with the 
effects of impressed forces of the harmonic type, and we may then 
advantageously employ the more special formulae applicable to such 
forces. In using normal co-ordinates, we have first to calculate the 
forces <&i, 4>,, &C. corresponding to each period, and thence deduce 
the values of the co-ordinates themselves. If among the natural 
periods (calculated without allowance for friction) there be any 
nearly agreeing in magnitude with the period of an impressed 
force, the corresponding component vibrations will be abnormally 
large, unless indeed the force itself be greatly attenuated in the 
preliminary resolution. Suppose, for example, that a transverse 
force of harmonic type and given period acts at a single point of 
a stretched string. All the normal modes of vibration will, in 
general, be excited, not however in their own proper periods, but 



99.] EQUILIBRIUM THEORY. 133 

in the period of the impressed force ; but any normal component, 
which has a node at the point of application, will not be excited. 
The magnitude of each component thus depends on two things : 
(1) on the situation of its nodes with respect to the point at which 
the force is applied, and (2) on the degree of agreement between 
its own proper period and that of the force. It is important to 
remember that in response to a simple harmonic force, the system 
will vibrate in general in all its modes, although in particular 
cases it may sometimes be sufficient to attend to only one of them 
as being of paramount importance. 

100. When the periods of the forces operating are very long 
relatively to the free periods of the system, an equilibrium theory 
is sometimes adequate, but in such a case the solution could 
generally be found more easily without the use of the normal 
co-ordinates. Bernoulli's theory of the Tides is of this class, and 
proceeds on the assumption that the free periods of the masses of 
water found on the globe are small relatively to the periods of the 
operative forces, in which case the inertia of the water might be 
left out of account. As a matter of fact this supposition is only 
very roughly and partially applicable, and we are consequently 
still in the dark on many important points relating to the tides. 
The principal forces have a semi-diurnal period, which is not suffi- 
ciently long in relation to the natural periods concerned, to allow 
of the inertia of the water being neglected. But if the rotation of 
the earth had been much slower, the equilibrium theory of the 
tides might have been adequate. 

A corrected equilibrium theory is sometimes useful, when the 
period of the impressed force is sufficiently long in comparison 
with most of the natural periods of a system, but not so in the 
case of one or two of them. It will be sufficient to take the case 
where there is no friction. In the equation 

a4> + C(f> = <P, or ^-f n*^ = */a, 

suppose that the impressed force varies as cos pt Then 

^ = 4>-T-a(n*-;)«) (1). 

The equilibrium theory neglects jp* in comparison with w*, 
and takes 

<^ = <E>^an« (2). 



134 VIBRATING SYSTEMS IN GENERAL. [100. 

Suppose now that this course is justifiable, except in respect 
of the single normal co-ordinate <^. We have then only to add 
to the result of the equilibrium theory, the difference between 
the true and the there assumed value of <^i, viz. 

ai(ni^—p*) OiW,' "" Til' — p' * a ^ ^' 

The other extreme case ought also to be noticed. If the 
forced vibrations be extremely rapid, they may become nearly 
independent of the potential energy of the system. Instead 
of neglecting p" in comparison with n', we have then to neglect 
7i« in comparison with p», which gives 

<^ = -4)^ap^ (4). 

If there be one or two co-ordinates to which this treatment 
is not applicable, we may supplement the result, calculated on 
the hypothesis that V is altogether negligible, with corrections 
for these particular co-ordinates. 

101. Before passing on to the general theory of the vibrations 
of systems subject to dissipation, it may be well to point out 
some peculiarities of the free vibrations of continuous systems, 
started by a force applied at a single point. On the suppositions 
and notations of § 93, the configuration at any time is deter- 
mined by 

?=^l^ + <^2^2+^8^+ (1), 

where the normal co-ordinates satisfy equations of the form 

Suppose now that the system is held at rest by a force applied 
at the point Q. The value of 4>r is determined by the considera- 
tion that 4>rS<^ represents the work done upon the system by the 
impressed forces during a hypothetical displacement B^=S<f>rUr, 
that is 

S<l>rjZurdx; 

thus ^r—IZtLrdiC=^Ur{Q) jZd^'y 

so that initially by (2) 

Cr(f>r = 'Ur{Q)lzdx (3). 



101.] SPECIAL INITIAL CONDITIONS. 135 

If the system be let go from this configuration at ^ = 0, we 
have at any subsequent time t, 

Vr{Q)fzdx Ur{Q)jZdx 

(f>r=OOBnrt =COSM j. (4), 

^ fir* IpV^^dx 

and at the point P 

Ur(P)Ur(Q)JZdx 

f=2cosnr^ (5). 

rir^jpUr^dx 

At particular points tLr(P) and Ur(Q) vanish, but on the 
whole 



Ur(P)Vr(Q)-^jpyrdx 



neither converges, nor diverges, with r. The series for f therefore 
converges with nr~\ 

Again, suppose that the system is started by an impulse 
from the configuration of equilibrium. In this case initially 

ar4>r = l^rdt = Ur (Q) jZidx, 

whence at time t 



* 



r^ —- -UriQ). Z,dx=^ ^^^IZ^dx (6). 

This gives 

Vr{P)Ur{Q)\z,dx 

f^Ssinn^ (7), 

nr \pu^da> 

shewing that in this case the series converges with 71^*"^ that 
is more slowly than in the previous case. 

In both cases it may be observed that the value of f is 
symmetrical with respect to P and Q, proving that the displace- 
ment at time t for the point P when the force or impulse is ap- 
plied at Q, is the same as it would be at Q if the force or impulse 
had been applied at P. This is an example of a very general 
reciprocal theorem, which we shall consider at length presently. 



136 VIBRATING SYSTEMS IN GENERAL. [101. 

As a third case we may suppose the body to start fix)m rest 
as deformed by a force uniformly distributed, over its length, 
area, or volume. We readily find 

Ur{P).Z.^u4a: 

^=%co^nrb — ■ (8). 

Wy* \pu^dx 

The series for f will be more convergent than when the force 
is concentrated in a single point. 

In exactly the same way we may treat the case of a con- 
tinuous body whose motion is subject to dissipation, provided 
that the three functions T, F, V be simultaneously reducible, 
but it is not necessary to write down the formulae. 

102. If the three mechanical functions T, F and V of any 
system be not simultaneously reducible, the natural vibrations 
(as has already been observed) are more complicated in their 
character. When, however, the dissipation is small, the method 
of reduction is still useful ; and this class of cases besides being 
of some importance in itself will form a good introduction to 
the more general theory. We suppose then that T and V are 
expressed as sums of squares 

F=ici^i' + ica<^,» + ...j ^ ^' 

while F still appears in the more general form 

F = ^bn4>i' + hb„4>,' + ...-^b^<f>,4>,+ (2). 

The equations of motion are accordingly 

ai4>i 4- 6ii^ + 6u^ + 6i,<^j + ... + Ci<l>i = I 

as4>« + 6n9i + &H<^2 + &a<^8+ ••• +Ca^s = / (3), 

in which the coefficients 6u» &u, &c. are to be treated as small. 
If there were no friction, the above system of equations would 
be satisfied by supposing one co-ordinate <^r to vary suitably, 
while the other co-ordinates vanish. In the actual case there 
will be a corresponding solution in which the value of any other 
co-ordinate <f>t will be small relatively to (f>r. 

Hence, if we omit terms of the second order, the r^ equation 
becomes, 

Or^r 4- 6r»^r + Cr^r = (4), 



102.] SMALL DI8SIPAT1VE FOBCES. 137 

from which we infer that <f>r varies approximately as if there 
were no change due to friction in the type of vibration. If 4>r 
vary as e^, we obtain to determine pr 

ttrPr* + itrPr 4- Cr = (5). 

The roots of this equation are complex, but the real part 
is small in comparison with the imaginary part. [The character 
of the motion represented by (5) has already been discussed 
(§ 45). The rate at which the vibrations die down is proportional 
to brr, and the period, if the term be still admitted, is approxi- 
mately the same as if there were no dissipation.] 

From the s^ equation, if we introduce the supposition that 
all the co-ordinates vary as e^, we get 

terms of the second order being omitted ; whence 

^ . ^ - ^ri»j>r h$Pr /^x 

'^' ■ '^'--J7^::^r'c^^^^) ^^^- 

This equation determines approximately the altered type 
of vibration. Since the chief part of pr is imaginary, we see 
that the co-ordinates <^« are approximately in the same phase, 
hit that that phase differs hy a quarter period from the phase 
of <f>r. Hence when the function F does not reduce to a sum 
of squares, the character of the elementary modes of vibration 
is less simple than otherwise, and the various parts of the system 
are no longer simultaneously in the same phase. 

We proved above that, when the friction is small, the value 
of Pr may be calculated approximately without allowance for 
the change of type ; but by means of (6) we may obtain a still 
closer approximation, in which the squares of the small quantities 
are retained. The r^^ equation (3) gives 

arp/^Cr^b„pr + t^^^^^=0 (7). 

The leading part of the terms included under S being real, 
the correction has no effect on the real part of pr on which 
the rate of decay depends. 

102 a. Following the electrical analogy we may conveniently 
describe the forces expressed by F as forces of resistance. In 
§ 102 we have seen that if the resistances be small, the periods 
are independent of them. We may therefore extend to this case 



138 VIBRATING SYSTEMS IN GENERAL. [l02 a. 

the application of the theorems with regard to the eflfect upon 
the periods of additions to T and F, which have been already 
proved when there are no resistances. 

By (5) § 102, if the forces of resistance be increased, the rates 
of subsidence of all the normal motions are in general increased 
with them; but in particular cases it may happen that there 
is no change in a rate of subsidence. 

It is natural to inquire whether this conclusion is limited to 
small resistances, for at first sight it would appear likely to hold 
good generally. An argument sufficient to decide this question 
may be founded upon a particular case. Consider a system formed 
by attaching two loads at any points of a stretched string vibrating 
transversely. If the mass of the string itself be neglected, there 
are two degrees of freedom and two periods of vibration corre- 
sponding to two normal modes. In each of these modes both loads 
in general vibrate. Now suppose that a force of resistance is 
introduced retarding the motion of one of the loads, and that this 
force gradually increases. At first the effect is to cause both kinds 
of vibration to die out and that at an increasing rate, but after- 
wards the law changes. For when the resistance becomes infinite, 
it is equivalent to a constraint, holding at rest the load upon which 
it acts. The remaining vibration is then unaffected by resistance, 
and maintains itself indefinitely. Thus the rate of subsidence of 
one of the normal modes has decreased to evanescence in spite of a 
continual increase in the forces of resistance F. This case is of 
course sufficient to disprove the suggested general theorem. 

103. We now return to the consideration of the general 
equations of § 84. 

If V^ii V^a» &c. be the co-ordinates and 'V^^'V^, &c. the forces, 
we have 

eflV^i + «aV^2+...='«'j,&c.J ^ ^' 

where c„ = a„Z)» 4- 6r#-D 4- c^, (2). 

For the fi-ee vibrations ^i, &c. vanish. If V be the de- 
terminant 

v^ e^ f ^, (o) 



103.] GENERAL EQUATIONS. 139 

the result of eliminating from (1) all the co-ordinates but one, is 

VVr = (4). 

Since V now contains odd powers of Z), the 2m roots of the 
equation V = no longer occur in equal positive and negative 
pairs, but contain a real as well as an imaginary part. The 
complete integral may however still be written 

yjr^A^^^ + A'ef'^'^ + Be^ + B'e^'^+ (5), 

where the pairs of conjugate roots are Ah, /^' ; fi^t fh '> &c. Corre- 
sponding to each root, there is a pai*ticular solution such as 

in which the ratios Ai : A^ : A^,,, are determined by the equa- 
tions of motion, and only the absolute value remains arbitrary. 
In the present case however (where V contains odd powers of D) 
these ratios are not in general real, and therefore the variations 
of the co-ordinates >^i, yjr^, &c. are not sjmchronous in phase. If 

we put /ii = ai + i)8i» A^' = «i— */9i» &c., we see that none of the 
quantities a can be positive, since in that case the energy of 
the motion would increase with the time, as we know it cannot 
do. 

103 a. The general argument (§§ 85, 103) from considerations 
of energy as to the nature of the roots of the determinantal 
equation (Thomson and Tait's Natural Philosophy, 1st edition 1867) 
has been put into a more mathematical form by Ilouth\ His 
investigation relates to the most general form of the equation in 
which the relations § 82 

Ctr« = a«., hrt^bsTf Crn = Ctr (1), 

are not assumed. But for the sake of brevity and as sufficient 
for almost all acoustical problems, these relations will here be 
supposed to hold. 

We shall have occasion to consider two solutions corresponding 
to two roots /i, V of the equation. For the first we have 

Vr,=Jtf,e^S V^, = J^f,e'*^>8 = i^^3e^^&c (2), 

and for the second 

^,^N,^, ir,=^N,e^, ir,^N,e^,&,c (3). 

In either of these solutions, for example (2), the ratios 

Ml : M^i Mi : 

^ Rigid Dynamict, 5th edition, Ch. vii. 



140 VIBRATING SYSTEMS IN GENERAL. [103 a. 

are determinate when ^ has been chosen. They are real when 
/i is real ; and when /a is complex (a + iyS), they take the form 

P ± iQ. 

If now we substitute the values of '^ from (2) in the equations 

of motion, we get 

(Oii/A* + hufji, + Cii) ifi + (oiiAt* 4- 6is/iA + Ci,) if, + = \ 

(a^ + bi^ + Cu) JIfi + (a^ + 6jaAt + CM)if2+ = p--W- 

The first result is obtained by multiplying these equations in 
order by Mi, if,, &c. and adding. It may be written 

^/i« + £/i+C=0, (5), 

where 

il = iOxiif," + iOaif,' + Oij^fiif, + (6). 

B = ^bnMi^ + i6aif,« + 6ijKJfiif,+ (7). 

C = ic„ifi* + JCjaJf,' + Ci5^1fiif,+ (8). 

The functions A, jB, C, are, it will be seen, the same as we have 
already denoted by T, F, and V respectively; but the varied 
notation may be useful as reminding us that there is as yet no 
limitation upon the nature of these quadratic functions. 

The following inferences from (5) are drawn by Routh : — 
(a) If Ay ByC either be zero, or be one-signed functions of 

the same sign, the fundamental determinant cannot have a real 

positive root. For if /a were real, the coeflBcients ifi, if,, 

would be real. We should thus have the sum of three positive 

quantities equal to zero. 

(fi) If there be no forces of resistance, i.e. if the term B be 

absent, and if A and C be one-signed and have the same sign, 

the fundamental determinant cannot have a real root, positive or 

negative. 

(7) If ^, jB, be one-signed functions, but if the sign of 

B be opposite to that of A and (7, the fundamental determinant 

cannot have a real negative root. 

The second equation is obtained as before from (4), except that 

now the multipliers are iVj, iV,,... appropriate to the root 1/. The 

result may be written 

il(/i,i/)/i«+5(M,i')/i + C(M,i/) = (9), 

where 

2A (/JL, v) = auMiNi 4- ojkf^ar, + 

+ ai,(M,N,+ M,NO+ (10), 



103 a.] bouth's thbobems. 141 

with similar suppositions for B{fjL,v) and C(fi,v). A(fjL,p) is 
thus a symmetrical function of the M's and UTa, so that 

A(ji,p) = A(p,,i) (11). 

It will be observed that according to this notation A (fi, fi) is 
the same as ^ in (6). 

In like manner 

A(ji,v)v' + B(fi,v)v + C(fi,v)=^0 (12), 

shewing that fi, v are both roots of the quadratic, whose* co- 
efficients are A (/a, i/), B (ji, v), C (/a, v). Accordingly 

^ + i; = --_P — - /Ai'^-jT — ; (13). 

We will now suppose that /a, p are two conjugate complex 

roots, so that 

/i = a + i^, 1/ = a — 1)8, 

where a, /3 are real. Under these circumstances if J/i, if,, ... be 

Pi + iQi, i'. + iQ,,..., then N^.N,,... will be Pi-iQu P,-iQ,y 
, the Fs and Q's being real. Thus by (10) 

2^(/i,i;) = a,,(P,«+Qx')-fa«(P,^+Q,«) + 

+ 2a^(P,P, + Q,Q,)+ 

= 2^(P) + 2^(Q) (14). 

In (14) A(P)t A(Q) are functions, such as (6), of real variables. 

From (13) we now find 

B(P) + B(Q) 

^-~A(P)~+MQ) ^^^>' 

„,. o,_C(P) + C7(Q) 

" -^^-AJPHMQ) ^ ^' 

From these Routh deduces the following conclusions : — 

(S) If A and B be one-signed and have the same sign 
(whether Che a, one-signed function or not), then the real part a 
of every imaginary root must be negative and not zero. But if B 
be absent, then the real part of every imaginary root is zero. 

(e) If A and C be one-signed and have opposite signs, then 
whatever may be the character of B, there can be no imaginary 
roots. 

It may be remarked that if jS do not occur, and if /i' and v* 
be diflferent roots of the determinant, it follows from (9), (12) that 

A{,jL.p)^CifL,p)^0 (17). 



142 VIBBATING SYSTEMS IN GENERAL. [103 a. 

When the number of degrees of freedom is finite, the funda- 
mental determinant may be expanded in powers of /a, giving 
an equation / (/i) =0 of degree 2m. The condition of stability 
is that all the real roots and the real parts of all the complex 
roots should be negative. If, as usual, complex quantities x + iy 
be represented by points whose co-ordinates are x, y, the condition 
is that all points representing roots should lie to the left of the 
axis of y. The application of Cauchy*s rule relative to the 
number of roots within any contour, by taking as the contour the 
infinite semi-circle on the positive side of the axis of y, is veiy 
fiiUy discussed by RouthS who has thrown the results into forms 
convenient for practical application to particular cases. 

103 6. The theorems of § 103 a do not exhaust all that general 
mechanical principles would lead us to expect as to the character 
of the roots of the fundamental determinant, and it may be well 
to pursue the question a little further. We will suppose through- 
out that A is one-signed and positive. 

If B and C be both one-signed and positive, we see that the 
equilibrium is thoroughly stable ; for from (a) it follows that there 
can be no positive root, and from (S) that no complex root can have 
its real part positive. 

In like manner the equations of § 103 a suffice for the case 
where C is one-signed and positive, B one-signed and negative. 
By (5) every real root is positive, and by (15) the real part 
of every complex root. Hence the equilibrium is unstable in 
every mode. 

When G is one-signed and negative, all the roots are real (S) ; 
but (5) does not tell us whether they are positive or negative. 
When jB = 0, we know (§ 87) that the roots occur in pairs of equal 
numerical value and of opposite sign. In this case therefore 
there are m positive and m negative roots. We vdll prove that 
this state of things cannot be disturbed by B. For if the determi- 
nant be expanded, the coefficient of fi^ is the discriminant of -4, 
and the coefficient of fi^ is the discriminant of C, By supposition 
neither of these quantities is zero, and thus no root of the equation 
can be other than finite. Hence as B increases from zero to its 
actual magnitude as a function of the variables, no root of the 
equation can change sign, and accordingly there remain iu 

^ Adams Prize Essay 1S77 ; Rigid Dynamics § 290. 



1 3 &. ] INSTABILITY. 143 

positive and m negative roots. It should be noticed that in this 
argument there is no restriction upon the character of B, 

In the case of a real root the values of M^ if,, ... are real, and 
thus the motion is such as might take place under a constraint 
reducing the system to one degree of freedom. But if this con- 
straint were actually imposed, there would be two corresponding 
values of /i, being the values given by (5). In general only one of 
these is applicable to the question in hand. Otherwise it would 
be possible to define m kinds of constraint, one or other of which 
would be consistent with any of the 2m roots. But this could 
only happen when the three functions A, B, G are simultaneously 
reducible to sums of squares (§ 97). 

When jB = 0, there are ni modes of motion, and two roots for 
each mode. In the present application to the case where C is 
one-signed and negative, each of the m modes for 5=0 gives 
one positive and one negative root. The positive root denotes 
instability, and although the negative root gives a motion which 
diminishes without limit, the character of instability is considered 
to attach to the mode as a whole, and all the m modes are said 
to be unstable. But when B is finite, there are in general 2m 
distinct modes with one root corresponding to each. Of the 
2m modes m are unstable, but the remaining m modes must be 
reckoned as stable. On the whole, however, the equilibrium is 
unstable, so that the influence of B, even when positive, is in- 
sufficient to obviate the instability due to the character of C. 

We must not prolong much further our discussion of unstable 
systems, but there is one theorem respecting real roots too 
fundamental to be passed over. It may be regarded as an ex- 
tension of that of § 88. 

The value of fi corresponding to a given constraint M^iM^: ,,, 
is one of the roots of (5) : and it follows from (4) that the value of 
fi is stationary when the imposed constraint coincides with one of 
the modes of free motion. The effect of small changes in A, B, G 
may thus be calculated from (5) without allowance for the 
accompanying change of type. 

Let (7, being negative for the mode under consideration, 
undergo numerical increase, while A and B remain unchanged as 
functions of the co-ordinates. The latter condition requires that 
the roots of (5), one of which is positive and one negative, should 
move either both towards zero or both away fi-om zero ; and the 
first condition excludes the former alternative. Whether it be 



144 VIBRATING SYSTEMS IN GENERAL. [103 6. 

the positive or the negative root of (5) which is the root of the 
determiDant, we infer that the change in question causes the 
latter to move away from zero. 

In like manner if A increase, while B and C remain unchanged, 
the movement of the root, whether positive or negative, is 
necessarily towards zero. 

Again, if A and C be given, while B increases algebraically 
as a function of the variables, the movement of the root of the 
determinant must be in the positive direction. 

Ad algebraic increase in B thus increases the stability, or 
decreases the iostability, in every mode. A numerical increase 
in C or decrease in A on the other hand promotes the stability 
of the stable modes and the instability of the unstable modes. 

We can do little more than allude to the theorem relating to 
the eflfect of a single constraiut upon a system for which C is 
one-signed and negative. Whatever be the nature of 5, the 
(m— 1) positive roots of the determinant, appropriate to the 
system after the constraint has been applied, will separate the m 
positive roots of the original determiuant, and a like proposition 
will hold for the negative roots. Upon this we may found a 
generalization of the foregoing conclusions analogous to that 
of § 92 a. Consider an independent vibrator of one degree of 
freedom for which C is positive, and let the roots of the frequency 
equation be ViyV^^ one negative and one positive. If we regard 
this as forming part of the system, we have in all (2m + 2) roots. 
The eflFect of a constraint by which the two parts of the system 
are connected will be to reduce the (2m + 2) back to 2m. Of 
these the m positive will separate the (m -f 1) quantities formed 
of the m positive roots of the original equation together with (the 
positive) 1/,, and a similar proposition will hold for the negative 
roots. The eflfect of the vibrator upon the original system is thus 
to cause a movement of the positive roots towards 1/3, and a 
movement of the negative roots towards i/j. This conclusion 
covers all the previous statements as to the eflfect of changes in 
A, B,C upon the values of the roots. 

Enough has now been said on the subject of the free vibra- 
tions of a system in general. Any further illustration that it 
may require will be aflforded by the discussion of th^ case of two 
degrees of freedom, § 112, and by the vibrations of strings and other 
special bodies with which we shall soon be occupied. We resume 



103 6.] FORCED VIBBATI0N8. 145 

the equations (1) § 103, with the view of investigating further the 
nature oi forced vibrations. 

104. In order to eliminate from the equations all the co- 
ordinates but one ('^i), operate on them in succession with the 
minor determinants 

dV dV dV ^^ 
den * de^ ' de^ * 

and add the results together; and in like manner for the other 
co-ordinatea We thus obtain as the equivalent of the original 
system of equations 

V I -^ Mr ^ Mr ^^ \sr 
^^ den ^ de^ ' de^i * 



\JbV\g llrC22 ^"^^tlt 

de^g cfejg d^a 



(1). 



in which the differentiations of V are to be made without re- 
cognition of the equality subsisting between ert &nd etr- 

The forces "^^j, ^^, &c. are any whatever, subject, of course, 
to the condition of not producing so great a displacement or 
motion that the squares of the small quantities become sensible. 
If, as is often the case, the forces operating be made up of two 
parts, one constant with respect to time, and the other periodic, 
it is convenient to separate in imagination the two classes of 
effects produced. The effect due to the constant forces is exactly 
the same as if they acted alone, and is found by the solution 
of a statical problem. It will therefore generally be sufficient 
to suppose the forces periodic, the effects of any constant forces, 
such as gravity, being merely to alter the configuration about 
which the vibrations proper are executed. We may thus without 
any real loss of generality confine ourselves to periodic, and 
therefore by Fourier's theorem to harmonic forces. 

We might therefore assume as expressions for "9^, &c. circular 
functions of the time; but, as we shall have frequent occasion 
to recognise in the course of this work, it is usually more con- 
venient to employ an imaginary exponential function, such as 
E(?^^, where £ is a constant which may be complex. When the 



146 VIBRATING SYSTEMS IN GENERAL. [104. 

corresponding symbolical solution is obtained, its real and 
imaginary parts may be separated, and belong respectively to 
the real and imaginary parts of the data. In this way the 
analysis gains considerably in brevity, inasmuch as differentiations 
and alterations of phase are expressed by merely modifying the 
complex coefficient without changing the form of the function. 
We therefore write 

^i = ^ie»i^, "ir^^E^e^P^, &c. 

dV 
The minor determinants of the type -j— are rational integral 

(l€r$ 

functions of the symbol D, and operate on ^i, &c. according to 
the law 

f{D)^p^^f(ip)e^ (2). 

Our equations therefore assume the form 

Vi/ri = ^ie*^, Vi/r, = ^e»lrt, &c (3), 

where -4i, -4,, &c. are certain complex constants. And the sym- 
bolical solutions are 

or by (2), Vri = ^^, &c (4), 

where V (ip) denotes the result of substituting ip for D in V. 

Consider first the case of a system exempt from friction. 

V and its differential coefficients are then even functions of 
D, so that V(ip) is real. Throwing away the imaginary part 
of the solution, writing iJie**> for -4i, &c., we have 

ti = v^cos(;)^ + ^0, &c (5). 

If we suppose that the forces '*'i, &c. (in the case of more 
than one generalized component) have all the same phase, they 
may be expressed by 

J&iC0s(jt)i4-a), j^a cos ( jt)^ -f a), &c. ; 

and then, as is easily seen, the co-ordinates themselves agree 
in phase with the forces: 

ti = v^-)Cos(pe + a) (6). 

The amplitudes of the vibrations depend among other things 
on the magnitude of V(ijt)). Now, if the period of the forces 



104.] FORCED VIBRATIONS. 147 

be the same as one of those belonging to the free vibrations, 
V (ip) = 0, and the amplitude becomes infinite. This is, of 
course, just the case in which it is essential to introduce the 
consideration of friction, from which no natural system is really 
exempt. 

If there be friction, V (ip) is complex ; but it may be divided 
into two parts — one real and the other purely imaginary, of which 
the latter depends entirely on the friction. Thus, if we put 

V(ip) = V,(tp) + i>V,(ip) (7), 

Vi, Vj are even functions of ip, and therefore real. If as before 
A I = RiC^', our solution takes the form 

or, on throwing away the imaginary part, 

^ iZiCosO)<-f gi-fy) ,g. 

where tan7=:- ^^^*^^ (9). 

We have said that V, (ip) depends entirely on the friction ; but 
it is not true, on the other hand, that V^ {ip) is exactly the same, 
as if there had been no friction. However, this is approximately 
the case, if the friction be small ; because any part of V (ip), which 
depends on the first power of the coefficients of friction, is neces- 
sarily imaginary. Whenever there is a coincidence between the 
period of the force and that of one of the free vibrations, Vj {ip) 
vanishes, and we have tan 7 = — oo , and therefore 

_i?,s in(p^ + gi) . . 

^^" i>V,(tp) <^^^' 

indicating a vibration of large amplitude, only limited by the 
friction. 

On the hypothesis of small friction, is in general small, and 
so also is 7, except in case of approximate equality of periods. 
With certain exceptions, therefore, the motion has nearly the 
same (or opposite) phase with the force that excites it. 

When a force expressed by a harmonic term acts on a system, 
the resulting motion is everywhere harmonic, and retains the 
original period, provided always that the squares of the displace- 



148 VIBRATING SYSTEMS IN GENERAL. [104. 

ments and velocities may be neglected. This important principle 
was enunciated by Laplace and applied by him to the theory of 
the tides. Its great generality was also recognised by Sir John 
Herschel, to whom we owe a formal demonstration of its truths 

If the force be not a harmonic function of the time, the types 
of vibration in different parts of the system are in general different 
from each other and from that of the force. The harmonic 
functions are thus the only ones which preserve their type un- 
changed, which, as was remarked in the Introduction, is a strong 
reason for anticipating that they correspond to simple tones. 

106. We now turn to a somewhat different kind of forced 
vibration, where, instead of given /orce« as hitherto, given inexora- 
ble motions are prescribed. 

If we suppose that the co-ordinates -^i, -^j, ... -^^ are given 
functions of the time, while the forces of the remaining types 
"^r+i, "^r+ai ••• ^m vauish, the equations of motion divide them- 
selves into two groups, viz. 

and 

(2). 

In each of the m— r equations of the latter group, the first r 
terms are known explicit functions of the time, and have the same 
effect as known forces acting on the system. The equations of 
this group are therefore suflScient to determine the unknown 
quantities ; after which, if required, the forces necessary to main- 
tain the prescribed motion may be determined from the fii*st 
group. It is obvious that there is no essential difference between 
the two classes of problems of forced vibrations. 

106. The motion of a system devoid of friction and executing 
simple harmonic vibrations in consequence of prescribed variations 
of some of the co-ordinates, possesses a peculiarity parallel to those 
considered in §§ 74, 79. Let 

'^JTi^ Ai cos pty '^, = -4,cos/>^, &c., 

^ Encyc, Metrop, art. S28. Also Outlines ofAttronomy, § 650. 



106.] INEXORABLE MOTIONS. 149 

in which the quantities Ai...Ar are regarded as given, while the 
remaining ones are arbitrary. We have from the expressions for 
Tand F,§82, 

2(r+F) = i(cu + i)'a,0^i' + ... + (Ci,+i>*aia)^i^-f-... 

+ {h (cii -l>* Ou) -4j' + . .. + (cu -p'Oij) AiAi + . ..} cos 2pt, 
from which we see that the equations of motion express the con- 
dition that E, the variable part of T + F, which is proportional to 

^{Cu- P^CLn)Ai*+ ... -^(Ca- p^dii) AiA^-^- (1), 

shall be stationary in value, for all variations of the quantities 
-4^+1 ... -4m. Let jt)'' be the value of |>* natural to the system when 
vibrating under the restraint defined by the ratios 

then 

8o that 

E^ip'^-p") {^il,« + ... +auiliil, + ...} (2). 

From this we see that if p* be certainly less than p*^ ; that is, 
if the prescribed period be greater than any of those natural to 
the system under the partial constraint represented by 

then E is necessarily positive, and the stationary value — there can 
be but one — is an absolute minimum. For a similar reason, if the 
prescribed period be leas than any of those natural to the partially 
constrained system, £ is an absolute maximum algebraically, but 
arithmetically an absolute minimum. But when p lies within the 
range of possible values of jt)'*, E may be positive or negative, and 
the actual value is not the greatest or least possible. Whenever 
a natural vibration is consistent with the imposed conditions, that 
will be the vibration assumed. The variable part of r+ F is then 
zero. 

For convenience of treatment we have considered apart the 
two great classes of forced vibrations and free vibrations; but there 
is, of course, nothing to prevent their coexistence. After the lapse 
of a sufficient interval of time, the free vibrations always dis- 
appear, however small the friction may be. The case of abso- 
lutely no friction is purely ideal. 

There is one caution, however, which may not be superfluous 
in respect to the case where given motions are forced on the 
system. Suppose, as before, that the co-ordinates -^i, '^2»"«'^r are 
given. Then the free vibrations, whose existence or non-existence 



150 VIBRATING SYSTEMS IN GENERAL. [106. 

is a matter of indifference so far as the forced motion is concerned, 
must be understood to be such as the system is capable of, when 
the co-ordinates V^i-.V^r ctre not allowed to vary from zero. In 
order to prevent their varying, forces of the corresponding, types 
must be introduced ; so that from one point of view the motion in 
question may be regarded as forced. But the applied forces are 
merely of the nature of a constraint ; and their effect is the same 
as a limitation on the freedom of the motion. 

106 a. The principles of the last sections shew that if 
"^i* '^a«««'^r be given harmonic functions of the time Aicoapt, 
A^cospt,.,., the forces of the other tjrpes vanishing, then the 
motion is determinate, unless p is ao chosen as to coincide with 
one of the values proper to the system when -^i, -^j.-.-^r are 
maintained at zero. As an example, consider the case of a 
membrane capable of vibrating transversely. If the displacement 
-^ at every point of the contour be given (proportional to cos pt), 
then in general the value in the interior is determinate ; but an 
exception occurs if p have one of the values proper to the 
membrane when vibrating with the contour held at rest. This 
problem is considered by M. Duhem^ on the basis of a special 
analytical investigation by Schwartz. It will be seen that it may 
be regarded as a particular case of a vastly more general theorem. 

A like result may be stated for an elastic solid of which the 
surface motion (proportional to cos pt) is given at every point. Of 
course, the motion at the boundary need not be more than partially 
given. Thus for a mass of air we may suppose given the motion 
luyrmal to a closed surface. The internal motion is then deter- 
minate, unless the frequency chosen is one of those proper to the 
mass, when the surface is made unyielding. 

107. Very remarkable reciprocal relations exist between the 
forces and motions of different types, which may be regarded as 
extensions of the corresponding theorems for systems in which 
only F or T has to be considered (§ 72 and §§ 77, 78). If we sup- 
pose that all the component forces, except two — ^^i and '9^ — are 
zero, we obtain from § 104, 



cteii de^ 

dci2 * de^s 

^ Coun de Phynque Mathimatique, Tome Second, p. 190, Paris, 1891. 



(!)• 



107.] RECIPROCAL THEOREM. 151 

We now consider two cases of motion for the same system ; first 
when "^j vanishes, and secondly (with dashed letters) when '9i 
vanishes. If ^j = 0, 

t.=v-'£^. <2>- 

Similarly, if %' = 0, 

^•'=^-£^'' <»)• 

In these equations V and its differential coefficients are rational 
integral functions of the symbol £>', and since in every case 
^^ = ^^, V is a symmetrical determinant, and therefore 

d^ ^dV 



'n wow 



Hence we see that if a force "^^i act on the system, the co- 
ordinate '^2 is related to it in the same way as the co-ordinate '^/ 
is related to the force '^./, when this latter force is supposed to act 
alone. 

In addition to the motion here contemplated, there may be 
free vibrations dependent on a disturbance already existing at the 
moment subsequent to which all new sources of disturbance are 
included in '9 ; but these vibrations are themselves the effect of 
forces which acted previously. However small the dissipation 
may be, there must be an interval of time after which free vibra- 
tions die out, and beyond which it is unnecessary to go in taking 
account of the forces which have acted on a system. If therefore 
we include under '9 forces of sufficient remoteness, there are no 
independent vibrations to be considered, and in this way the 
theorem may be extended to cases which would not at first sight 
appear to come within its scope. Suppose, for example, that the 
system is at rest in its position of equilibrium, and then begins to 
be acted on by a force of the first type, gradually increasing in 
magnitude from zero to a finite value "^^i, at which point it ceases 
to increase. If now at a given epoch of time the force be sud- 
denly destroyed and remain zero ever afterwards, free vibrations of 
the system will set in, and continue until destroyed by friction. 
At any time t subsequent to the given epoch, the co-ordinate -^j 
has a value dependent upon t proportional to "^^i. The theorem 
allows us to assert that this value '^s beai*s the same relation to '9i 
as '^JTi would at the same moment have borne to '9^', if the original 
cause of the vibrations had been a force of the second type in- 



152 VIBRATING SYSTEMS IN GENERAL. [107. 

creasing gradually from zero tx> ^/, and then suddenly vanishing 
at the given epoch of time. We have already had an example of 
this in § 101, and a like result obtains when the cause of the 
original disturbance is an impulse, or, as in the problem of the 
pianoforte-string, a variable force of finite though short duration. 
In these applications of our theorem we obtain results relating to 
free vibrations, considered as the residual effect of forces whose 
actual operation may have been long before. 

108. In an important class of cases the forces ^i and '^Z are 
harmonic, and of the same period. We may represent them by 
A^ef^, ^jV^^ where A^ and A^ may be assumed to be real, if the 
forces be in the same phase at the moments compared. The 
results may then be written 

where ip is written for i). Thus, 

^>,= il,Vr/ (2). 

Since the ratio ^j : ^,' is by hypothesis real, the same is 

true of the ratio '^i : -^j; which signifies that the motions 

represented by those symbols are in the same phase. Passing 
to real quantities we may state the theorem thus: — 

If a force ^i = Aj cos pt, acting on the system give rise to 
the motion '^a = ^AiCos(pt — e); then mil a force "^^j' = Aj' cos pt 
produce the motion yjri = ^ Aj' cos (pt — e). 

If there be no friction, e will be zero. 

If Ai^Ai\ then '^/ = '^2« But it must be remembered that 
the forces "^^i and ^/ are not necessarily comparable, any more 
than the co-ordinates of corresponding types, one of which for 
example may represent a linear and another an angular dis- 
placement. 

The reciprocal theorem may be stated in several ways, but 
before proceeding to these we will give another investigation, 
not requiring a knowledge of determinants. 

If *i,^„... V^,,-^,,... and ^/, ^/,... Vr/, i/r,',... be two sets 



108.] RECIPROCAL THEOREM. 153 

of forces and corresponding displacements, the equations of 
motion, § 103, give 

+ ^a' (6,1 Vri + eai/ra + tfai/r, + ...) + ... . 

Now, if all the forces vary as e**, the eflfect of a symbolic 
operator such as en on any of the quantities '^ is merely to 
multiply that quantity by the constant found by substituting 
ip for D in er«. Supposing this substitution made, and having 
regard to the relations ert='egr, we may write 

+ «i2(ti>a + t.>i)+ (3). 

Hence by the symmetry 

^it/ + ^>>/^/+- = ^i>i + *.>«+ W' 

which is the expression of the reciprocal relation. 

109. In the applications that we are about to make it 
will be supposed throughout that the forces of all types but 
two (which we may as well take as the first and second) are 
zero. Thus 

^iti' + ^»t/ = ^i>i + *>V» 0). 

The consequences of this equation may be exhibited in three 
different ways. In the first we suppose that 

whence ^/r, : ^i = >/r/ : ^,' (2), 

shewing, as before, that the relation of -^a to "^i in the first 
case when '^^2 = is the same as the relation of yjri to "^^/in 
the second case, when "^^jsO, the identity of relationship ex- 
tending to phase as well as amplitude. 

A few examples may promote the comprehension of a law, 
whose extreme generality is not unlikely to convey an impression 
of vagueness. 

If P and Q be two points of a horizontal bar supported in 
any manner (e.g. with one end clamped and the other free), a 
given harmonic transverse force applied at P will give at any 
moment the same vertical deflection at Q as would have been 
found at P, had the force acted at Q. 

If we take angular instead of linear displacements, the 



154 VIBRATING SYSTEMS IN GENERAL. [l09. 

theorem will run: — A given harmonic couple at P will give the 
same rotoHon at Q as the couple at Q would give at P. 

Or if one displacement be linear and the other angular, the 
result may be stated thus: Suppose for the first case that a 
harmonic couple acts at P, and for the second that a vertical 
force of the same period and phase acts at Q, then the linear 
displacement at Q in the first case has at every moment the 
same phase as the rotatory displacement at P in the second, 
and the amplitudes of the two displacements are so related that 
the maximum couple at P would do the same work in acting 
over the maximum rotation at P due to the force at Q, as the 
maximum force at Q would do in acting through the maximum 
displacement at Q due to the couple at P. In this case the 
statement is more complicated, as the forces, being of diflferent 
kinds, cannot be taken equal. 

If we suppose the period of the forces to be excessively long, 
the momentary position of the system tends to coincide with 
that in which it would be maintained at rest by the then acting 
forces, and the equilibrium theory becomes applicable. Our 
theorem then reduces to the statical one proved in § 72. 

As a second example, suppose that in a space occupied by 
air, and either wholly, or partly, confined by solid boundaries, 
there are two spheres A and B, whose centres have one degree 
of freedom. Then a periodic force acting on A will produce 
the same motion in B, as if the parts were interchanged ; and ' 
this, whatever membranes, strings, forks on resonance cases, or 
other bodies capable of being set into vibration, may be present in 
their neighbourhood. 

Or, if A and B denote two points of a solid elastic body 
of any shape, a force parallel to OX, acting at A, will produce 
the same motion of the point B parallel to OF as an equal force 
parallel to OY acting at B would produce in the point A, 
parallel to OX. 

Or again, let A and B be two points of a space occupied by 
air, between which are situated obstacles of any kind. Then a 
sound originating at A is perceived at B with the same intensity 
as that with which an equal sound originating at B would be per- 
ceived at -4.^ The obstacle, for instance, might consist of a rigid 

^ Helmholtz, CreUe, Bd. ltii., 1859. The sounds must be such as in the absence 
of obstacles woold diffuse themselves equaUy in all directions. 



109.] APPLICATIONS. 155 

wall pierced with one or more holes. This example corresponds 
to the optical law that if by any combination of reflecting or 
refracting surfaces one point can be seen from a second, the second 
can also be seen from the first. In Acoustics the sound shadows 
are usually only partial in consequence of the not insignificant 
value of the wave-length in comparison with the dimensions of 
ordinary obstacles: and the reciprocal relation is of considerable 
interest. 

A further example may be taken from electricity. Let there 
be two circuits of insulated wire A and B, and in their neigh- 
bourhood any combination of wire-circuits or solid conductors 
in communication with condensers. A periodic electro-motive 
force in the circuit A will give rise to the same current in B 
as would be excited in A if the electro-motive force operated- 
in B. 

Our last example will be taken from the theory of conduction 
and radiation of heat, Newton's law of cooling being assumed 
as a basis. The temperature at any point 2I of a conducting and 
radiating system due to a steady (or harmonic) source of heat 
at jB is the same as the temperature at B due to an equal source 
at A. Moreover, if at any time the soui*ce at B be removed, the 
whole subsequent course of temperature at A will be the same as 
it would be at B if the parts of B and A were interchanged. 

110. The second way of stating the reciprocal theorem is 
arrived at by taking in (1) of § 109, 

>/^i = 0, i^; = 0; 

whence ^iV^/ = ^'a>, (1)» 

or ^, : yfr,^^,' : i/r/ (2), 

shewing that the relation of ^1 to -^a in the first case, when yjri = 0, 
is the same as the relation of ^j' to yjri in the second case, when 

Thus in the example of the rod, if the point P be held at 
rest while a given vibration is imposed upon Q (by a force there 
applied), the reaction at P is the same both in amplitude and 
phase as it would be at Q if that point were held at rest and 
the given vibration were imposed upon P, 

So if A and B be two electric circuits in the neighbourhood 
of any number of others, C, D,... whether closed or terminating 



156 VIBRATING SYSTEMS IN GENERAL. [UO. 

in condensers, and a given periodic current be excited in A by 
the necessary electro-motive force, the induced electro-motive 
force in £ is the same as it would be in il, if the parts of A 
and B were interchanged. 

The third form of statement is obtained by putting in (I) 

of § 109, 

^1 = 0, Vr/ = 0; 

whence ^/V^i + ^a>i = (3), 

or t, : >;r, = -^/ : >P/ (4), 

proving that the ratio of -^i to -^j in the first case, when ^^ acts 
alone, is the negative of the ratio of ^i' to '^'x' ^^ the second case, 
when the forces are so related as to keep '^/ equal to zero. 

Thus if the point P of the rod be held at rest while a periodic 
force acts at Q, the reaction at P bears the same numerical ratio 
to the force at Q as the displacement at Q would bear to the 
displacement at P, if the rod were caused to vibrate by a force 
applied at P. 

111. The reciprocal theorem has been proved for all systems 
in which the frictional forces can be represented by the function F, 
but it is susceptible of a further and an important generalization. 
We have indeed proved the existence of the function F for 
a large class of cases where the motion is resisted by forces 
proportional to the absolute or relative velocities, but there are 
other sources of dissipation not to be brought under this head, 
whose effects it is equally important to include; for example, the 
dissipation due to the conduction or radiation of heat. Now 
although it be true that the forces in these cases are not for all 
possible motions in a constant ratio to the velocities or displace- 
ments, yet in any actual case of periodic motion (t) they are 
necessarily periodic, and therefore, whatever their phase, ex- 
pressible by a sum of two terms, ooe proportional to the dis- 
placement (absolute or relative) and the other proportional to the 
velocity of the part of the system aflFected. If the coeflScients 
be the same, not necessarily for all motions whatever, but for all 
motions of the period t, the function F exists in the only sense 
required for our present purpose. In fact since it is exclusively 
with motions of period r that the theorem is concerned, it is 
plainly a matter of indifference whether the functions T, F, V 
are dependent upon r or not. Thus extended, the theorem is 



111.] BECIPROCAL THEOREM. 157 

perhaps sufficiently geneml to cover the whole field of dissipative 
forces. 

It is important to remember that the Principle of Reciprocity 
is limited to systems which vibrate about a configuration of equi- 
librium, and is therefore not to be applied without reservation to 
such a problem as that presented by the transmission of sonorous 
waves through the atmosphere when disturbed by wind. The 
vibrations must also be of such a character that the square of the 
motion can be neglected throughout; otherwise our demonstra- 
tion would not hold good. Other apparent exceptions depend on 
a misunderstanding of the principle itself. Care must be taken 
to observe a proper correspondence between the forces and dis- 
placements, the rule being that the action of the force over the 
displacement is to represent tuork done. Thus couples correspond 
to rotations, pressures to increments of volume, and so on. 

Ill a. The substance of the preceding sections is taken from 
a paper by the Author *, in which the action of dissipative forces 
appears first to have been included. Reciprocal theorems of a 
special character, and with exclusion of dissipation, had been 
previously given by other writers. One, due to von Helmholtz, 
has already been quoted. Reference may also be made to the 
reciprocal theorem of Betti', relating to a uniform isotropic elastic 
solid, upon which bodily and surface forces act. Lamb^ has shewn 
that these results and more recent ones of von Helmholtz* may 
be deduced from a very general equation established by Lagrange 
in the Micanique Analytique. 

Ill 6. In many cases of practical interest the external force, 
in response to which a system vibrates harmonically, is applied at a 
single point. This may be called the driving-point, and it becomes 
important to estimate the reaction of the system upon it. When 
T and F only are sensible, or F and V only, certain general 
conclusions may be stated, of which a specimen will here be given. 
For further details reference must be made to a paper by the 
Author*. 

^ ** Some General Theorems relating to Vibrations," Proe, Math. Soc,, 1S73. 

s II Nuovo Cimento, 1S72. 

» Proe. Math. Soc., Vol. xix., p. 144, Jan. 18SS. 

* CreUe, t. 100, pp. 187, 213. 1886. 

* '*The Reaction upon the Driving-point of a System executing Forced Harmonic 
Oscillations of various Periods.*' Phil. Mag., May,>1886. 



158 VIBRATING SYSTEMS IN GENERAL. [ill 6. 

Consider a system, devoid of potential energy, in which the 
co-ordinate -^i is made to vary by the operation of the harmonic 
force "^^i , proportional to ff^. The other co-ordinates may be chosen 
arbitrarily, and it will be very convenient to choose them so that 
no product of them enters into the expressions for 2^ and F, They 
would be in fact the normal co-ordinates of the system on the 
supposition that ^^ is constrained (by a suitable force of its own 
type) to remain zero. The expressions for T and F thus take the 
following forms : — 

T = iani^i^ + \a^^^ + \(h,'^} + . . . 

-h aia>/^,>/r, + (hz^x<t% + (hA^ii^A + (1). 

• * • • * 

The equations for a force '9i, proportional to ^^, are accordingly 
(t>iii + 6ii) ^1 + (t^Ois 4- 6ij) ^, + (ipoi, + 615) >^5 -h . . . = '^^i , 
(ipoi, + 61,) ^i + (ipcha + M -^2 = 0, 



By means of the second and following equations -^j, "^s ••• are 
expressed in terms of ^1. Introducing these values into the first 
equation, we get 

The ratio '^^i/'^i is a complex quantity, of which the real part 
corresponds to the work done by the force in a complete period 
and dissipated in the system. By an extension of electrical 
language we may call it the resistance of the system and denote it 
by the letter R\ The other part of the ratio is imaginary. If we 
denote it by ipL'yjri^ or i'^i, L' will be the moment of inertia, or 
self-induction of electrical theory. We write therefore 

>P, = (iZ' + tpi') ^1 W; 

and the values of R' and Z' are to be deduced by separation of the 
real and the imaginary parts of the right-hand member of (3). In 
this way we get 

This is the value of the resistance as determined by the 
constitution of the system, and by the frequency of the imposed 



1116.] REACTION AT DRIVING-POINT. 159 

vibration. Each component of the latter series (which alone 
involves/)) is of the form ap^/ifi + 7p*), where a, /8, y are all positive, 
and (as may be seen most easily by considering its reciprocal) 
increases continually as jj^ increases from zero to infinity. We 
conclude that as the frequency of vibration increases, the value of 
R increases continuously with it. At the lower limit the motion 
is determined sensibly by the quantities b (the resistances) only, and 
the corresponding resultant resistance Bf is an absolute minimum, 
whose value is 

6„-2(6„VM (6). 

At the upper limit the motion is determined by the inei*tia of 
the component parts without regard to resistances, and the value 
of R is 

Ou-^-j-^^ — r^n » 

On 0^(hi' 



6n + 2f6„^;-26,^-i?) (7). 



or 

When p is either very large or very small, all the co-ordinates 
are in the same phase, and (6), (7) may be identified with 

Also U^a,,-'i'^ + 'i^'^^^?y^'Xx (8). 

In the latter series every term is positive, and continually 
diminishes as p^ increases. Hence every increase of frequency is 
attended by a diminution of the moment of inertia, which tends 
ultimately to the minimum corresponding to the disappearance of 
the dissipative terms. 

If /) be either very large or very small, (8) identifies itself 
with 2TI^^\ 

As a simple example take the problem of the reaction upon 
the primary circuit of the electric currents generated in a neigh- 
bouring secondary circuit. In this ciise the co-ordinates (or rather 
their rates of increase) are naturally taken to be the currents 
themselves, so that -^i is the primary, and ^^ the secondary 
current. In usual electrical notation we represent the coefficients 
of self-induction by i, iV, and of mutual induction by itf, so that 

and the resistances by R and S, Thus 

aii = L, Oij = if, Oa = iV ; 
ill = -Ri 6i3 = 0, iaj = iS ; 



160 VIBRATING SYSTEMS IN GENERAL. [ill fe. 

and (5) and (8) become at once 

*-«-^> «■ 

These formul» were given originally by Maxwell, who remarked 
that the reaction of the currents in the secondary has the effect 
of increasing the effective resistance and diminishing the effective 
self-induction of the primary circuit. 

If the rate of alternation be very slow, the secondary circuit is 
without influence. If, on the other hand, the rate be very rapid, 

112. In Chapter ill. we considered the vibrations of a system 
with one degree of freedom. The remainder of the present Chapter 
will be devoted to some details of the case where the degrees of 
freedom are two. 

If X and y denote the two co-ordinates, the expressions for T 
and V are of the form 

2F=i4a:«+25a?y + Cy^ J ^^' 

so that, in the absence of friction, the equations of motion are 

Lx + My + Ax^By^X\ . . 

Mx + Ny + Bx+Cy^Y] ^^^• 

When there are no impressed forces, we have for the natural 

vibrations 

(LI> -\-A)x + (MD' + B)y^O\ 

(MD^'^B)x+{ND'-\'C)y = 0' 

D being the symbol of differentiation with respect to time. 

If a solution of (3) be x = l€^, y=^m€^^, X' is one of the 

roots of 

{L\^-\-A)(N\^-\-C)-(MV + By = (4), 

or 

\^(LN-M^) + \^(LC'\-NA-2MB) + AC'-B' = (5). 

The constants L, M, N\ A, B, C, are not entirely arbitrary. 

Since T and V are essentially positive, the following inequalities 

must be satisfied : — 

LN>M\ AG>B' (6). 

Moreover, L, N, A, C must themselves be positive. 



(3). 



112.] ROOTS OP DETERMINANTAL EQUATION. 161 

We proceed to examine the eflFect of these restrictions on the 
roots of (5). 

In the first place the three coefficients in the equation are 
positive. For the first and third, this is obvious from (6). The 
coefficient of X^ 

- {JLG -jNAy 4- 2jLNAC - 2MB, 



in which, as is seen from (6), JLNAC is necessarily greater than 
MB. We conclude that the values of X', if real, are both 
negative. 

It remains to prove that the roots are in fact real. The 

condition to be satisfied is that the following quantity be not 

negative : — 

{LC^NA-2MBf-^{LN-M^){AC-B'). 

After reduction this may be brought into the form 

^{jLN.B^jAC.My 
+ (JLC^ J'NAy {{JLC - JNA )« + 4 {jLNAC - MB)}, 



which shews that the condition is satisfied, since JLNAC — MB 
is positive. This is the analytical proof that the values of X* are 
both real and negative ; a fact that might have been anticipated 
without any analysis from the physical constitution of the system, 
whose vibrations they serve to express. 

The two values of X' are different, unless both 

Jln.b-Jag.m^o 

JLC^JNA^O 
which require that 

L : M : N = A : B : C (7). 

The common spherical pendulum is an example of this case. 

By means of a suitable force Y the co-ordinate y may be 
prevented from varying. The system then loses one degree of 
freedom, and the period corresponding to the remaining one is in 
general difierent from either of those possible before the introduc- 
tion of Y, Suppose that the types of the motions obtained by 
thus preventing in turn the variation of y and x are respectively 
e^»^, e^. Then fii^, ^i^ are the roots of the equation 

(iX^ + il)(iV'X«-|-(7) = 0, 

R. \\ 



162 VIBRATING SYSTEMS IN GENERAL. [112. 

being that obtained from (4) by suppressing M and B, Hence 
(4) may itself be put into the form 

LN (k'-' fii')(\'- fi^)--(M\' + By (8), 

which shews at once that neither of the roots of X* can be 
intermediate in value between fii^ and /ij*. A little further 
examination will prove that one of the roots is greater than both 
the quantities /L^l^ ^*, and the other less than both. For if we put 

f(V) = LN (X» - /^») (X» - M.') - (if V + B)\ 
we see that when X^ is very small, / is positive (AC — B*); when 
X' decreases (algebraically) to fjLi\ f changes sign and becomes 
negative. Between and y^ there is therefore a root ; and also 
by similar reasoning between \t^ and — oo . We conclude that the 
tones obtained by subjecting the system to the two kinds of con- 
straint in question are both intermediate in pitch between the 
tones given by the natural vibrations of the system. In particular 
cases /Lti', /ia' may be equal, and then 

jTNji-±B ^^jA C±B 
J'LNTM JLN + M ^ ^' 

This proposition may be generalized. Any kind of constraint 
which leaves the system still in possession of one degree of free- 
dom may be regarded as the imposition of a forced relation 
between the co-ordinates, such as 

ax + $y = (10). 

Now if ow? + /8y, and any other homogeneous linear func- 
tion of X and y, be taken as new variables, the same argument 
proves that the single period possible to the system after the 
introduction of the constraint, is intermediate in value between 
those two in which the natural vibrations were previously per- 
formed. Conversely, the two periods which become possible 
when a constraint is removed, lie one on each side of the original 
period. 

If the values of X' be equal, which can only happen when 

L : M : N^A : B : C, 

the introduction of a constraint has no effect on the period ; for 
instance, the limitation of a spherical pendulum to one vertical 
plane. 

113. As a simple example of a system with two degrees of 
freedom, we may take a stretched string of length 2, itself without 



113.] 



FREE VIBRATIONS. 



163 



inertia, but carrying two equal masses m at distances a and h from 
one end (Fig. 17). Tension = 2\. 

Fig. 17. 



If X and y denote the displacements, 



f 



c^rr m [^ (^ - V) V 



Since T and V are not of the same form, it follows that the 
two periods of vibration are in every case unequal. 

If the loads be symmetrically attached, the character of the 
two component vibrations is evident. In the firat, which will have 
the longer period, the two weights move together, so that x. and y 
remain equal throughout the vibration. In the second x and y are 
numerically equal, but opposed in sign. The middle point of the 
string then remains at rest, and the two masses are always to 
be found on a straight line passing through it. In the first case 
a; — y = 0, and in the second x-k-y — Q] so that x — y and x + y 
ai-e the new variables which must be assumed in order to reduce 
the functions Tand V simultaneously to a sum of squares. 

For example, if the masses be so attached as to divide the 
string into three equal parts. 



2T = J' [{x^.yy + {x-yr] ] 



2 

Q/77 

2F=^' {(^+y)'+3(a;-y)'j 



(1). 



from which we obtain as the complete solution, 

a: + y = 4cos(y/?^\e 



+ a 



\ 



x-y = Bco.(J\^\t^^)^ 



(2), 



where, as usual, the constants A, a, By jS are to be determined by 
the initial circumstances. 

114. When the two natural periods of a system are nearly 
equal, the phenomenon of intermittent vibration sometimes pre- 
sents itself in a very curious manner. In order to illustrate this, 



164 VIBRATING SYSTEMS IN GENERAL. [114. 

we may recur to the string loaded, we will now suppose, with two 
equal masses at distances from its ends equal to one-fourth of the 
length. If the middle point of the string were absolutely fixed, 
the two similar systems on either side of it would be completely 
independent, or, if the whole be considered as one S3rstem, the two 
periods of vibration would be equal. We now suppose that 
instead of being absolutely fixed, the middle point is attached to 
springs, or other machinery, destitute of inertia, so that it is 
capable of yielding slightly. The reservation as to inertia is to 
avoid the introduction of a third degree of freedom. 

From the symmetry it is evident that the fundamental vibra- 
tions of the system are those represented hy x-\-y and x — y. 
Their periods are slightly diflferent, because, on account of the 
yielding of the centre, the potential energy of a displacement 
when X and y are equal, is less than that of a displacement 
when X and y are opposite; whereas the kinetic energies are 
the same for the two kinds of vibration. In the solution 

x + y = ^cos(nie + a) | 

j?-y=£cos(w,^ + /8) J ^ ^' 

we are therefore to regard tij and n^ as nearly, but not quite, equal. 
Now let us suppose that initially x and x vanish. The conditions 
are 

A cos a + £ cos /8 = 
7ii-4 sin a + n^B sin /8 = j ' 

which give approximately 

Thus a:^A^m'^^t sin f'^^^e + a) ) 

^ ^ ^ ^ \ (2). 

y = Aco6 — t cos f ^^ ^ "^ * ) ) 

The value of the co-ordinate x is here approximately ex- 
pressed by a harmonic term, whose amplitude, being proportional 
to sin ^ (w, - th) ^, is a slowly varying harmonic function of the 
time. The vibrations of the co-ordinates are therefore intermittent, 
and so adjusted that each amplitude vanishes at the moment that 
the other is at its maximum. 

This phenomenon may be prettily shewn by a tuning fork of 
very low pitch, heavily weighted at the ends, and firmly held by 



114.] INTERMITTENT VIBRATIONS. 165 

screwing the stalk into a massive support. When the fork vibrates 
in. the normal manner, the rigidity, or want of rigidity, of the 
stalk does not come into play ; but if the displacements of the two 
prongs be in the same direction, the slight yielding of the stalk 
entails a small change of period. If the fork be excited by striking 
one prong, the vibmtions are intermittent, and appear to transfer 
themselves backwards and forwards between the prongs. Unless, 
however, the support be very firm, the abnormal vibration, which 
involves a motion of the centre of inertia, is soon dissipated ; and 
then, of course, the vibration appears to become steady. If the 
fork be merely held in the hand, the phenomenon of intermittence 
cannot be obtained at all. 

116. The stretched string with two attached masses may be 
used to illustrate some general principles. For example, the period 
of the vibration which remains possible when one mass is held 
at rest, is intermediate between the two free periods. Any in- 
crease in either load depresses the pitch of both the natural 
vibrations, and conversely. If the new load be situated at a point 
of the string not coinciding with the places where the other loads 
are attached, nor with the node of one of the two previously 
possible free vibrations (the other has no node), the effect is still 
to prolong both the periods already present. With regard to the 
third finite period, which becomes possible for the first time after 
the addition of the new load, it must be regarded as derived from 
one of infinitely small magnitude, of which an indefinite number 
may be supposed to form part of the system. It is instructive 
to trace the effect of the introduction of a new load and its gradual 
increase from zero to infinity, but for this purpose it will be 
simpler to take the case where there is but one other. At the 
commencement there is one finite period Ti, and another of in- 
finitesimal magnitude r,. As the load increases r^ becomes finite, 
and both Ti and T] continually increase. Let us now consider 
what happens when the load becomes very great. One of the 
periods is necessarily large and capable of growing beyond all 
limit. The other must approach a fixed finite limit. The first 
belongs to a motion in which the larger mass vibrates nearly as 
if the other were absent ; the second is the period of the vibration 
of the smaller mass, taking place much as if the larger were fixed. 
Now since t^ and t, can never be equal, Ti must be always the 
greater ; and we infer, that as the load becomes continually larger^ 



166 VIBRATING SYSTEMS IN G£N£BAL. [115. 

it is Ti that increases indefinitely, and r^ that approaches a finite 
limit. 

We now pass to the consideration of forced vibrations. 

116. The general equations for a system of two degrees of 
freedom including friction are 

In what follows we shall suppose that F = 0, and that X — e^^. 
The solution for y is 

If the connection between x and y be of a loose character, the 
constants Jf , /8, B are small, so that the term {B — p^M + ifipf 
in the denominator may in general be neglected. When this 
is permissible, the co-ordinate y is the same as if a? had been pre- 
vented from varying, and a force T had been introduced whose 
magnitude is independent of N, 7, and G, But if, in consequence 
of an approximate isochronism between the force and one of the 
motions which become possible when or or y is constrained to be 
zero, either A—p^L-^iap or C—p^N-\-iyp be small, then the 
term in the denominator containing the coefficients of mutual 
influence must be retained, being no longer relatively unimportant ; 
and the solution is accordingly of a more complicated character. 

Symmetry shews that if we had assumed X = Q, F =«*''*, we 
should have found the same value for x as now obtains for y. This 
is the Reciprocal Theorem of § 108 applied to a system capable 
of two independent motions. The string and two loads may again 
be referred to as an example. 

117. So far for an imposed force. We shall next suppose 
that it is a motion of one co-ordinate {x = 6*^) that is prescribed, 
while F=0; and for greater simplicity we shall confine ourselves 
to the case where )8 = 0. The value of y is 

y C-Np' + iyp ^ ^• 

Let us now inquire into the reaction of this motion on x. 
We have 



117.] REACTION OF A DEPENDENT SYSTEM. 167 

If the real and imaginary parts of the coefficient of e^^ be re- 
spectively A' and iap, we may put 

{MI> + B)y^A'x + a'x (3), 

and {B-Mi ^r{G-N tn (4) 

(C'-iVp»)« + 7«p« ^ ^' 

It appears that the effect of the reaction of y (over and above 
what would be caused by holding y = 0) is represented by changing 
A into A + A\ and a into a + a\ where A' and a' have the above 
values, and is therefore equivalent to the effect of an alteration in 
the coeflScients of spring and friction. These alterations, however, 
are not constants, but functions of the period of the motion con- 
templated, whose character we now proceed to consider. 

Let n be the value of jo corresponding to the natural frictionless 
period of y {x being maintained at zero); so that G—n^N — Q, 
Then 



«' = <^-^^)^^^-^4--^^J 



.(6). 



In most cases with which we are practically concerned 7 is 
small, and interest centres mainly on values of p not much differ- 
ing from n. We shall accordingly leave out of account the 
variations of the positive factor {B — Mp^f^ and in the small term 
7^, substitute for p its approximate value n. When p is not 
nearly equal to n, the term in question is of no importance. 

As might be anticipated from the general principle of work, 
a' is always positive. Its maximum value occurs when p = n 
nearly, and is then proportional to 1/771', which varies inversely 
with 7. This might not have been expected on a superficial view 
of the matter, for it seems rather a paradox that, the greater the 
friction, the less should be its result. But it must be remembered 
that 7 is only the coefficient of friction, and that when 7 is small 
the maximum motion is so much increased that the whole work 
spent against friction is greater than if 7 were more considerable. 

But the point of most interest is the dependence of A' on p, 
lip be less than n, ^' is negative. As p passes through the value 



168 VIBRATING SYSTEMS IN GENERAL. [117. 

n, A' vanishes, and changes sign. When A' is negative, the in- 
fluence of y is to diminish the recovering power of the vibration a?, 
and we see that this happens when the forced vibration is slower 
than that natural to y. The tendency of the vibration y is thus 
to retard the vibration a?, if the latter be already the slower, but 
to accelerate it, if it be already the more rapid, only vanishing in 
the critical case of perfect isochronism. The attempt to make x 
vibrate at the rate determined by n is beset with a peculiar 
difficulty, analogous to that met with in balancing a heavy 
body with the centre of gravity above the support. On which- 
ever side a slight departure from precisioi\^ of adjustment may 
occur the influence of the dependent vibiution is alwajrs to increase 
the error. Ebcamples of the instability of pitch accompanying a 
strong resonance will come across us hereafter ; but undoubtedly 
the most interesting application of the results of this section is to 
the explanation of the anomalous refraction, by substances possess- 
ing a very marked selective absorption, of the two kinds of light 
situated (in a normal spectrum) immediately on either side of the 
absorption band^ It was observed by Christiansen and Kundt, 
the discoverers of this remarkable phenomenon, that media of the 
kind in question (for example, /i^A^" we in alcoholic solution) refract 
the ray immediately hel(no the absorption-band abnormally in 
excess, and that above it in defect If we suppose, as on other 
grounds it would be natural to do, that the intense absorption is 
the result of an agreement between the vibrations of the kind of 
light affected, and some vibration proper to the molecules of the 
absorbing agent, our theory would indicate that for light of some- 
what greater period the eflfect must be the same as a relaxation of 
the natural elasticity of the ether, manifesting itself by a slower 
propagation and increased refraction. On the other side of the 
absorption-band its influence must be in the opposite direc- 
tion. 

In order to trace the law of connection between A' and p, take, 
for brevity, 7 n = a, i\r (p^ — n*) = a?, so that 

A 3C —- -. 

ic* + a^ 

When the sign of a; is changed. A' is reversed with it, but pre- 
serves its numerical value. When a? = 0, or ± 00 , -4' vanishes. 

^ Phil Mag., May, 1872. Also Sellmeier, Pogg, Ann. t. oxliii. p. 272, 1871. 



117.] 



BEACTION OF A DEPENDENT SYSTEM. 



169 




Hence the origin is on the representative curve (Fig. 18), and the 
axis of X is an asymptote. The maximum and minimum values of 
A' occur when x is respectively equal to + a, or — a ; and then 



X 



The corresponding values of jo are given by 



jp2 = n=± 



yn 

F 



(7). 



Hence, the smaller the value of a or 7, the greater will be the 
maximum alteration of Ay and the corresponding value of p will 
approach nearer and nearer to n. It may be well to repeat, that in 
the optical application a diminished 7 is attended by an increased 
maximum absorption. When the adjustment of periods is such as 
to favour A' as much as possible, the corresponding value of a' is 
one half of its maximum. 



CHAPTER VI. 

TRANSVERSE VIBRATIONS OF STRINGS. 

118. Among vibrating bodies there are none that occupy a 
more prominent position than Stretched Strings. From the 
earliest times they have been employed for musical purposes, 
and in the present day they still form the essential parts of such 
important instruments as the pianoforte and the violin. To the 
mathematician they must always possess a peculiar interest as the 
battle-field on which were fought out the controversies of DAlem- 
bert, Euler, Bernoulli and Lagrange, relating to the nature of the 
solutions of partial differential equations. To the student of 
Acoustics they are doubly important. In consequence of the com- 
parative simplicity of their theory, they are the ground on which 
difficult or doubtful questions, such as those relating to the nature 
of simple tones, can be most advantageously faced ; while in the 
form of a Monochord or Sonometer, they afford the most generally 
available means for the comparison of pitch. 

The 'string* of Acoustics is a perfectly uniform and flexible 
filament of solid matter stretched between two fixed points — in 
fact an ideal body, never actually realized in practice, though 
closely approximated to by most of the strings employed in music. 
We shall afterwards see how to take account of any small devia- 
tions from complete flexibility and uniformity. 

The vibrations of a string may be divided into two distinct 
classes, which are practically independent of one another, if the 
amplitudes do not exceed certain limits. In the first class the 
displacements and motions of the particles are longitudinal^ so 
that the string always retains its straightness. The potential 
energy of a displacement depends, not on the whole tension, but 
on the changes of tension which occur in the various parts of the 
string, due to the increased or diminished extension. In order to 



118.] TRANSVERSE VIBRATIONS OF STRINGS. 171 

calculate it we must know the relation between the extension of 
a string and the stretching force. The approximate law (given by 
Hooke) may be expressed by saying that the extension varies 
as the tension, so that if I and l denote the natural and the 
stretched lengths of a string, and T the tension, 

^---' = ^ (1) 

where ^ is a constant, depending on the material and the section, 
which may be interpreted to mean the tension that would be 
necessary to stretch the string to twice its natural length, if the 
law applied to so great extensions, which, in general, it is far 
firom doing. 

119. The vibrations of the second kind are transverse ; that is 
to say, the particles of the string move sensibly in planes perpen- 
dicular to the line of the string. In this case the potential energy 
of a displacement depends upon the general tension, and the 
small variations of tension accompanying the additional stretching 
due to the displacement may be left out of account. It is here 
assumed that the stretching due to the motion may be neglected 
in comparison with that to which the string is already subject in 
its position of equilibrium. Once assured of the fulfilment of this 
condition, we do not, in the investigation of transverse vibrations, 
require to know anything further of the law of extension. 

The most general vibration of the transverse, or lateral, kind 
may be resolved, as we shall presently prove, into two sets of 
component normal vibrations, executed in perpendicular planea 
Since it is only in the initial circumstances that there can be any 
distinction, pertinent to the question, between one plane and 
another, it is sufficient for most purposes to regard the motion as 
entirely confined to a single plane passing through the line of the 
string. 

In treating of the theory of strings it is usual to commence 
with two particular solutions of the partial diflferential equation, 
representing the transmission of waves in the positive and nega- 
tive directions, and to combine these in such a manner as to suit 
the case of a finite string, whose ends are maintained at rest ; 
neither of the solutions taken by itself being consistent with the 
existence of nodes, or places of permanent rest. This aspect of the 
question is very important, and we shall fully consider it ; but it 



172 TRANSVERSE VIBRATIONS OF STRINGS. [119. 

seems scarcely desirable to found the solution in the first instance 
on a property so peculiar to a uniform string as the undisturbed 
transmission of waves. We will proceed by the more general 
method of assuming (in conformity with what was proved in the 
last chapter) that the motion may be resolved into normal com- 
ponents of the harmonic tjrpe, and determining their periods and 
character by the special conditions of the system. 

Towards carrying out this design the first step would naturally 
be the investigation of the partial diflTerential equation, to which 
the motion of a continuous string is subject. But in order to 
throw light on a point, which it is most important to understand 
clearly, — the connection between finite and infinite freedom, and 
the passage corresponding thereto between arbitrary constants 
and arbitrary functions, we will commence by following a some- 
what diflTerent course. 

120. In Chapter iii. it was pointed out that the fundamental 
vibration of a string would not be entirely altered in character, 
if the mass were concentrated at the middle point. Following 
out this idea, we see that if the whole string were divided into a 
number of small parts and the mass of each concentrated at its 
centre, we might by suflSciently multiplying the number of parts 
arrive at a system, still of finite freedom, but capable of represent- 
ing the continuous string with any desired accuracy, so far at 
least as the lower component vibrations are concerned. If the 
analytical solution for any number of divisions can be obtained, 
its limit will give the result corresponding to a uniform string. 
This is the method followed by Lagrange. 

Let I be the length, pi the whole mass of the string, so that 
p denotes the mass per unit length, T^ the tension. 

Fig. 19. 



The length of the string is divided into m + 1 equal parts (a), 
so that 

(m + l)a = i (1). 



120.] MASS CONCENTRATED IN POINTS. 173 

At the m points of division equal masses (fi) are supposed con- 
centrated, which are the representatives of the mass of the por- 
tions (a) of the string, which they severally bisect. The mass of 
each terminal portion of length ^a is supposed to be concentrated 
at the final points. On this understanding, we have 

(m+l)fi = pl (2). 

We proceed to investigate the vibrations of a string, itself 
devoid of inertia, but loaded at each of vi points equidistant 
(a) from themselves and from the ends, with a mass fi. 

If >^i, >^j i^m+i denote the lateral displacements of the 

loaded points, including the initial and final points, we have the 
following expressions for T and V, 

y=iMl^i' + ^.* + ...+^m+i + ^m+a} (3) 

with the conditions that '^i and ^m+s vanisL These give by 
Lagrange's Method the m equations of motion. 



B^lr^ -\-Aylr:, +£>^4 =0 
Bylr^ +Aylr^ -^ Bylr, =0 

Bfm + Ayltm+i + Bylr^+, = 



(5), 



where A^iiD^^- \ £ = -ii (6). 

Supposing now that the vibration under consideration is one 
of normal t3rpe, we assume that >^i, "^j, &c. are all proportional to 
cos {nt — e), where n remains to be determined. A and B may 
then be regarded as constants, with a substitution of — ?i' for D*. 

If for the sake of brevity we put 

C=A^B = -2 + ^^^ (7), 

the determinantal equation, which gives the values of n^ assumes 
the form 



174 



TRANSVEBSE VIBRATIONS OF STRINGS. 



[120. 



a, 1, 0, 0, 0. 

1, C, 1, 0, 0. 

0, 1, G, 1, 0. 

0. 0, 1, G, I. 

0, 0, 0, 1, G. 



m rows 



= 0. 



(8). 



From this equation the values of the roots might be fouoA 
It may be proved that, if (7= 2 cos 6, the determinant is equivalent 
to sin (m + 1) ^ -r sin ; but we shall attain our object with greater 
ease directly from (5) by acting on a hint derived from the known 
results relating to a continuous string, and assuming for trial a 
particular type of vibration. Thus let a solution be 



'^^ = P 8in(r — 1))8 co8(n^ — e) 



(9). 



a form which secures that >/ri = 0. In order that -^m+j may 
vanish, 

(m+l)/3 = «7r (10), 

where « is an integer. Substituting the assumed values of ^/r in 
the equations (5), we find that they are satisfied, provided that 

2jBcos/3 + il=0 (11); 

so that the value of n in terms of )3 is 



"='^«-f\/S w 



A normal vibration is thus represented by 

J, . (r-l)sir , . 
Yr = Pi sin ^^ T — cos (n,< 



ftt + l 



-€.). 



where 



w, = 2a/-^ si 



sir 
fia 2(m4-l) 



(13). 



(14), 



and P„ €, denote arbitrary constants independent of the general 
constitution of the system. The m admissible values of n are 
found from (14) by ascribing to « in succession the values 1, 2, 
3...m, and are all diflferent. If we take s^m + l, -^^ vanishes, 
so that this does not correspond to a possible vibration. Greater 
values of s give only the same periods over again. If m H- 1 be 
even, one of the values of n — that, namely, corresponding to 



120.] MASS CONCENTRATED IN POINTS. 175 

5 = ^(m + l),— is the same as would be found in the case of only 
a single load (m = 1). The interpretation is obvious. In the kind 
of vibration considered every alternate particle remains at rest, so 
that the intermediate ones really move as though they were 
attached to the centres of strings of length 2a, fastened at 
the ends. 

The most general solution is found by putting together all the 
possible particular solutions of normal type 

irr^^l. P, sin ^ ^ ^ ^^, cos (n,t - €,) (15), 

♦■■I 7/lr -f- 1 

and, by ascribing suitable values to the arbitrary constants, can 
be identified with the vibration resulting from arbitrary initial 
circumstances. 

Let X denote the distance of the particle r from the end of the 
string, so that (r — l)a = a?; then by substituting for /x and a 
from (1) and (2), our solution may be written, 

■\lr{x) = P,sins J- coa(n,t — €,) (16), 

2(»H-1) /r; . s-TT 

"•=— r-\/7«^°2(m+i) ^i^>- 

In order to pass to the case of a continuous string, we have 
only to put m infinite. The first equation retains its form, and 
specifies the displacement at any point x. The limiting form of 
the second is simply 






(18), 

whence for the periodic time. 



-?-fy^ w 



The periods of the component tones are thus aliquot parts of 
that of the gravest of the series, found by putting 5 = 1. The 
whole motion is in all cases periodic ; and the period is 21 V0>/2\). 
This statement, however, must not be understood as excluding 
a shorter period; for in particular cases any number of the 
lower components may be absent. All that is asserted is that the 



176 



TRANSVERSE VIBRATIONS OF STRINGS. 



[120. 



above-mentioned interval of time is sufficient to bring about a com- 
plete recuri'ence. We defer for the present any further discussion 
of the important formula (19), but it is interesting to observe the 
approach to a limit in (17), as m is made successively greater and 
greater. For this purpose it will be sufficient to take the gravest 
tone for which 5 = 1, and accordingly to trace the variation of 
2 (m -h 1) 



sin 



TT 



TT 



2(m-f 1)' 



The following are a series of simultaneous values of the func- 
tion and variable : — 



m 


1 


2 


3 


4 


9 


19 


39 


2(m+l) . IT 
IT 2(m + l) 


•9003 


•9549 


•9745 


•9836 


•9959 


•9990 


•9997 



It will be seen that for very moderate values of m the limit is 
closely approached. Since m is the number of (moveable) loads, 
the case m= 1 corresponds to the problem investigated in Chapter 
III., but in comparing the results we must remember that we there 
supposed the whole mass of the string to be concentrated at the 
centre. In the present case the load at the centre is only half as 
great; the remainder being supposed concentrated at the ends, 
where it is without effect. 

From the fact that our solution is general, it follows that any 
initial form of the string can be represented by 



* = 00 



*rrx 



-^ (ic) = 2 _ (P cos €), sin 5 -^ 



(20). 



And, since any form possible for the string at all may be 
regarded as initial, we infer that any finite single valued function 
of X, which vanishes at a? = and x = l, can be expanded within 
those limits in a series of sines of irxjl and its multiples, — which 
is a case of Fourier's theorem. We shall presently shew how the 
more general form can be deduced. 

121. We might now determine the constants for a continuous 
string by integration as in § 93, but it is instructive to solve the 
problem first in the general case (m finite), and afterwards to 
proceed to the limit. The initial conditions are 



121.] MASS CJONCENTKATED IN POINTS. 177 

y(a) = -4i8in -T- H-ilj8m2-^ + ... 4--4„»8mm -^ , 

'i|r(2a) = ^1 sm 2 -T- + -4, sm 4 -p + . . . 4- .A^ sm 2m -^ , 



• / V -« • "^tt ^ • rt ^ct - . Tra 

y (ma) = -di8mm-T- +iljSin2m -v- +... +-4,«8in mm-j- ; 

where, for brevity, il, = P, cose,, and -^(a), '^(2a) '>^{ma) 

are the iDitial displacements of the m particles. 

To determine any constant -4,, multiply the first equation by 
8in(57ra/i), the second by sin (2«7ra//), &c., and add the results. 
Then, by Trigonometry, the coeflScients of all the constants, except 
A,y vanish, while that of ^, = ^ (m 4- 1)^ Hence 

^ — t2 "^^ira) sinr« -7- (1). 

m4-l»^i I 

We need not stay here to write down the values of jB, (equal 
to Pg sin €g) as depending on the initial velocities. When a becomes 
infinitely small, ra under the sign of summation ranges by infi- 

nitesimal steps from zero to L At the same time ^ = y , 

SO that writing ra = x, a^dx, we have ultimately 



-4, = yj ylr(x) sinf^jcte (2), 




expressing A, in terms of the initial displacements. 

122. We will now investigate independently the partial differ- 
ential equation governing the transverse motion of a perfectly 
flexible string, on the suppositions (1) that the magnitude of the 
tension may be considered constant, (2) that the square of the 
inclination of any part of the string to its initial direction may be 
neglected. As before, p denotes the linear density at any point, 
and Ti is the constant tension. Let rectangular co-ordinates be 
taken parallel, and perpendicular to the string, so that x gives the 
equilibrium and x, y, z the displaced position of any particle at 
time t The forces acting on the element dx are the tensions at 

I Todhunter's Int. Calc., p. 267. 
R. 1^ 



178 TRANSVERSE VIBRATIONS OF STRINGS. [122. 

its two ends, and any impressed forces Tp dx, Zp dx. By D'Alem- 
bert*8 Principle these form an equilibrating system with the 
reactions against acceleration, —pdh/ldt^y —pd^z/dt*. At the 
point X the components of tension are 

rp dy ^ dz 
^'dx' ^'di' 

if the squares of dy/dx, dzjdx be neglected ; so that the forces 
acting on the element dx arising out of the tension are 

Hence for the equations of motion, 



dt^ p da^^ 
dH Ti d^z ^ 



(1), 



dt^ p da^ 

from which it appears that the dependent variables y and z are 
altogether independent of one another. 

The student should compare these equations with the corre- 
sponding equations of finite diCFerences in § 120. The latter may 
be written 

/* 5^8 -^ W = — M*^ (^ - «) + '^ (^ + a) - 2i|r (a?)}. 

Now in the limit, when a becomes infinitely small, 
>/r (a? - a) + i|r (a? H- a) - 2'i|r (a?) = >/r" (a?) a', 
while p» — pa\ and the equation assumes ultimately the form 

agreeing with (1). 

In like manner the limiting forms of (3) and (4) of § 120 are 

dx (2), 



-^m 



=*^./(i) 



''''''i, (3), 



which may also be proved directly. 



122.] DIFFERENTIAL EQUATIONS. 179 

The first is obvious from the definition of T. To prove the 
second, it is sufficient to notice that the potential energy in any 
configuration is the work required to produce the necessary 
stretching against the tension Ti. Reckoning from the configura- 
tion of equilibrium, we have 



^'^•}{i-'h-' 



and, so far as the third power of -~ , 



^-l = if^)' 



123. In most of the applications that we shall have to make, 
the density p is constant, there are no impressed forces, and the 
motion may be supposed to take place in one plane. We may 
then conveniently write 

7-- <■>• 

and the differential equation is expressed by 

d{aty da^ ^ ^' 

If we now assume that y varies as cos mat, our equatipn 
becomes 

S+"^'y=o <3), 

of which the most general solution is 

y = (-4 sin7;ia; + (7cosma?)cosma^ (4). 

This, however, is not the most general harmonic motion of 

the period in question. In order to obtain the latter, we must 

assume. 

y =2 y^ cos mat + ya sin m^t (5), 

where yi, y^ are functions of x, not necessarily the same. On 
substitution in (2) it appears that yi and y, are subject to equations 
of the form (3), so that finally 

y = {A sin mx + G cos mx) cos mat ] ,^. 

+ (B8inmx + Dcosm>x)smmatj 

an expression containing four arbitrary constants. For any con- 
tinuous length of string satisfpng without interruption the differ- 

Vi— '^ 



180 TRANSVERSE VIBRATIONS OF STRINGS. [l23. 

ential equation, this is the most general solution possible, under 

the condition that the motion at every point .shall be simple 

harmonic. But whenever the string forms part of a system 

vibrating freely and without dissipation, we know from former 

chapters that all parts are simultaneously in the same phase, 

which requires that 

A : B = C : D (7); 

and then the most general vibration of simple harmonic type is 

y = {asin7n^ + )8cos mx} co8(mat — €) (8). 

124. The most simple as well as the most important problem 
connected with our present subject is the investigation of the free 
vibrations of a finite string of length I held fast at both its ends. 
If we take the origin of x at one end, the terminal conditions are 
that when x = 0, and when x = l, y vanishes for all values of t. 
The first requires that in (6) of § 123 

C=0, 2) = (1); 

and the second that 

8mml = (2), 

or that ml = sir, where s is an integer. We learn that the only 
harmonic vibrations possible are such as make 

^ = T (3). 

and then 

. STTX f . sirat „ . siratx 
y = sm-y-(-4 cos —, — hi5sm— v- ) (4). 

Now we know a priori that whatever the motion may be, it 
can be represented as a sum of simple harmonic vibrations, and 
we therefore conclude that the most general solution for a string, 
fixed at and I, is 

_•-• . STTX f . swat „ . 87rat\ 

The slowest vibration is that corresponding to 5 = 1. Its 
period (ti) is given by 






n=- = 2Z^/x, (6). 

The other components have periods which are aliquot parts 
of Ti : — 

T* = Ti-T-« (7); 



124.] FIXED EXTREMITIES. 181 

80 that, as has been already stated, the whole motion is under all 
circumstances periodic in the time Tj. Thp sound emitted con- 
stitutes in general a musical note, according to our definition of 
that term, whose pitch is fixed by Tj, the period of its gravest 
component. It may happen, however, in special cases that the 
gravest vibration is absent, and yet that the whole motion is not 
periodic in any shorter time. This condition of things occurs, if 
Ai^-^Bi* vanish, while, for example, A^-\-B^ and A^-^B^ are 
finite. In such cases the sound could hardly be called a note; 
but it usually happens in practice that, when the gravest tone is 
absent, some other takes its place in the character of fundamental, 
and the sound still constitutes a note in the ordinary sense, 
though, of course, of elevated pitch. A simple case is when all 
the odd components beginning with the first are missing. The 
whole motion is then periodic in the time ^Ti, and if the second 
component be present, the sound presents nothing unusual. 

The pitch of the note yielded by a string (6), and the character 
of the fundamental vibration, were first investigated on mechanical 
principles by Brook Taylor in 1715 ; but it is to Daniel Bernoulli 
(1755) that we owe the general solution contained in (5). He 
obtained it, as we have done, by the synthesis of particular 
solutions, permissible in accordance with his Principle of the 
Coexistence of Small Motions. In his time the generality of the 
result so arrived at was open to question ; in fact, it was the 
opinion of Euler, and also, strangely enough, of Lagrange ^ that 
the series of sines in (5) was not capable of representing an 
arbitrary function ; and Bernoulli's argument on the other side, 
drawn from the infinite number of the disposable constants, 
was certainly inadequate*. 

Most of the laws embodied in Taylor s formula (6) had been 
discovered experimentally long before (1636) by Mersenne. They 
may be stated thus : — 

1 See Riemann's Partielle Differential Gleiehungen, § 78. 

' Dr Toung, in his memoir of 1800, seems to have understood this matter quite 
correctly. He says, ** At the same time, as M. BemouUi has justly observed, since 
every figure may be infinitely approximated, by considering its ordinates as 
composed of the ordinates of an infinite number of trochoids of different magni- 
tudes, it may be demonstrated that all these constituent curves would revert to 
their initial state, in the same time that a similar chord bent into a trochoidal 
curve would perform a single vibration ; and this is in some respects a convenient 
and compendious method of considering the problem/' 



182 TRANSVERSE VIBRATIONS OF STRINGS. [l24. 

(1) For a given string and a given tension, the time varies as 
the length. 

This is the fundamental principle of the monochord, and 
appears to have been understood by the ancients \ 

(2) When the length of the string is given, the time varies 
inversely as the square root of the tension. 

(3) Strings of the same length and tension vibrate in times, 
which are proportional to the square roots of the linear density. 

These important results may all be obtained by the method of 
dimensions, if it be assumed that r depends only on I, p, and Ti. 

For, if the units of length, time and mass be denoted re- 
spectively by [L], [T], [if], the dimensions of these symbols are 
given by 

Z = [Z], p^[ML'^l T, = [MLl^l 

and thus (see § 52) the only combination of them capable of re- 
presenting a time is 2^"* .p^,l. The only thing left undetermined 
is the numerical &ctor. 

126. Mersenne's laws are exemplified in all stringed instru- 
ments. In playing the violin different notes are obtained from 
the same string by shortening its efficient length. In tuning 
the violin or the pianoforte, an adjustaient of pitch is effected 
with a constant length by varying the tension ; but it must be 
remembered that p is not quite invariable. 

To secure a prescribed pitch with a string of given material, it is 
requisite that one relation only be satisfied between the length, the 
thickness, and the tension; but in practice there is usually no great 
latitude. The length is often limited by considerations of con- 
venience, and its curtailment cannot always be compensated by 
an increase of thickness, because, if the tension be not increased 
proportionally to the section, there is a loss of flexibility, 
while if the tension be so increased, nothing is effected towards 
lowering the pitch. The difficulty is avoided in the lower strings 
of the pianoforte and violin by the addition of a coil of fine wire, 
whose effect is to impart inertia without too much impairing 
flexibility. 

^ Aristotle ** knew that a pipe or a chord of doable length produced a sound of 
which the vibrations occupied a doable time ; and that the properties of concords 
depended on the proportions of the times occupied by the vibrations of the 
separate sounds." — Young's Lectures on Natural Philoeqphy, Vol. i. p. 404. 



125.] mersenne's laws. 183 

For quantitative investigations into the laws of strings, the 
sonometer is employed. By means of a weight hanging over a 
pulley, a catgut, or a metallic wire, is stretched across two bridges 
mounted on a resonance case. A moveable bridge, whose position 
is estimated by a scale running parallel to the wire, gives the 
means of shortening the eflScient portion of the wire to any 
desired extent. The vibrations may be excited by plucking, as 
in the harp, or with a bow (well supplied with rosin), as in the 
violin. 

If the moveable bridge be placed half-way between the fixed 
ones, the note is raised an octave ; when the string is reduced to 
one-third, the note obtained is the twelfth. 

By means of the law of lengths, Mersenne determined for the 
first time the frequencies of known musical notes. He adjusted the 
length of a string until its note was one of assured position in the 
musical scale, and then prolonged it under the same tension until 
the vibrations were slow enough to be counted. 

For experimental purposes it is convenient to have two, or 
more, strings mounted side by side, and to vary in turn their 
lengths, their masses, and the tensions to which they are subjected. 
Thus in order that two strings of equal length may jdeld the 
interval of the octave, their tensions must be in the ratio of 1 : 4, 
if the masses be the same ; or, if the tensions be the same, the 
masses must be in the reciprocal ratio. 

The sonometer is very useful for the numerical determination 
of pitch. By varjdng the tension, the string is tuned to unison 
with a fork, or other standard of known frequency, and then by 
adjustment of the moveable bridge, the length of the string is 
determined, which vibrates in unison with any note proposed for 
measurement. The law of lengths then gives the means of 
effecting the desired comparison of fi-equencies. 

Another application by Scheibler to the determination of 
absolute pitch is important. The principle is the same as that 
explained in Chapter ill., and the method depends on deducing 
the absolute pitch of two notes from a knowledge of both the 
ratio and the difference of their frequencies. The lengths of the 
sonometer string when in unison with a fork, and when giving with 
it four beats per second, are carefully measured. The ratio of the 



184 TRANS VEBSE VIBRATIONS OF STRINGS. [l25. 

lengths is the inverse ratio of the frequencies, and the diflference 
of the frequencies is four. From these data the absolute pitch of 
the fork can be calculated. 

The pitch of a string may be calculated also by Taylor's 
formula from the mechanical elements of the system, but 
great precautions are necessary to secure accuracy. The tension 
is produced by a weight, whose mass (expressed with the same 
unit as p) may be called P; so that Ti=gP, where g = 32% 
if the units of length and time be the foot and the second. In 
order to secure that the whole tension acts on the vibrating 
segment, no bridge must be interposed, a condition only to be 
satisfied by suspending the string vertically. After the weight is 
attached, a portion of the string is isolated by clamping it firmly 
at two points, and the length is measured. The mass of the unit 
of length p refers to the stretched state of the string, and may be 
found indirectly by observing the elongation due to a tension 
of the same order of magnitude as 7\, and calculating what 
would be produced by 2\ according to Hooke's law, and by 
weighing a known length of the string in its normal state. 
After the clamps have been secured great care is required to 
avoid fluctuations of temperature, which would seriously influence 
the tension. In this way Seebeck obtained very accurate results. 

126. When a string vibrates in its gravest normal mode, the 
excursion is at any moment proportional to sin (irx/l), increasing 
numerically from either end towards the centre ; no intermediate 
point of the string remains permanently at rest. But it is other- 
wise in the case of the higher normal components. Thus, if the 
vibration be of the mode expressed by 

. 87rx ( . STrat ^ • S7rat\ 
y = sm -^ ( Ag cos —, — h B, sm — p J , 

the excursion is proportional to sin (sttx/I), which vanishes at « — 1 
points, dividing the string into s equal parts. These points of no 
motion are called nodes, and may evidently be touched or held 
fast without in any way disturbing the vibration. The produc- 
tion of * harmonics ' by lightly touching the string at the points of 
aliquot division is a well-known resource of the violinist. All 
component modes are excluded which have not a node at the 
point touched ; so that, as regards pitch, the effect is the same as 
if the string were securely fastened there. 



127.] NORMAL MODES. 185 

127. The constants, which occur in the general value of y, 
§ 124, depend on the special circumstances of the vibration, and 
may be expressed in terms of the initial values of y and y. 

Putting < = 0, we find 

yo = 2,^i -4,sm ^-; yo= y 2,.i «jB,sm -y- (1). 



STTX 



Multiplying by sin— p, and integrating from to I, we obtain 

^• = 7 yosm-v-cte; 5« = -— I yosm-j-cto (2). 

These results exemplify Stokes* law, § 95 ; for that part of y, which 
depends on the initial velocities, is 

^»"« 2 . 9Trx , snat /*' . . sirx , 
y=^._i — sm -J- sm — i— I Vosm— j— cw', 

^ '"^TTCW I I Jo i 

and from this the part depending on initial displacements may 
be inferred, by differentiating with respect to the time, and 
substituting y© for y©. 

When the condition of the string at some one moment is 
thoroughly known, these formnlse allow us to calculate the 
motion for all subsequent time. For example, let the string be 
initially at rest, and so displaced that it forms two sides of a 
triangle. Then 5, = ; and 




. 2y ( f^ X . STTX J f^ I —X . STTX J I 

= ,^^6(f-6) "^° T <^>' 

on integration. 

We see that Ag vanishes, if sin (sirb/l) = 0, that is, if there be 
a node of the component in question situated at P, A more 
comprehensive view of the subject will be afforded by another 
mode of solution to be given presently. 



186 TRANSVERSE VIBRATIONS OF STRINGS. [l28. 

128. In the expression for y the coefficients of sin (sirx/l) are 
the normal co-ordinates of Chapters iv. and v. We will denote 
them therefore by 0«, so that the configuration and motion of the 
system at any instant are defined by the values of (f>g and <^« 
according to the equations 

, . Tra? . . . 27rx , . STTX 

y^<l>iQin-j- + <l>2 sin — p- -h . . . H- <^, sm -y- + . . . 

^ ^ 1^ (1). 

; . TTX ; . 27rX . ; . STTX 

y = 0i8in -y- + 92sm -j-+ ... + 9,sm-y— + ... 

We proceed to form the expressions for T and F, and thence 
to deduce the normal equations of vibration. 

For the kinetic energy, 



= ipj 2,^1 4>i' sm' -^- dx, 



the product of every pair of terms vanishing by the general 
property of normal co-ordinates. Hence 



T = ipl^,^, <!>»' (2). 

In like manner, 

=w-x;:r^*.' w 

These expressions do not presuppose any particular motion, either 
natural, or otherwise; but we may apply them to calculate the 
whole energy of a string vibratiug naturally, as follows : — ^If M 
be the whole mass of the string (pi), and its equivalent (a^p) be 
substituted for 2\, we find for the sum of the energies, 

or, in terms of Ag and J5, of § 126, 

T+V^ir^M.tl^, dLiL£t (5). 



(4). 



128.] young's theorem. 187 

If the motion be not confined to the plane of xy, we have 
merely to add the energy of the vibrations in the perpendicular 
plane. 

Lagrange's method gives immediately the equation of motion 



*-+(T)"*-i*- »■ 



which has been already considered in § 66. If ^o ^^^ 4^o ^ ^^^ 
initial values of ^ and ^, the general solution is 



, ; sin nt , 
<f>=^ (pQ h <f>Q cos nt 



+ ^j\inn(t--f)<S>dt' (7), 



Ipn, 

where n is written for sira/l. 

By definition 4>« is such that 4>« S<f), represents the work done 
by the impressed forces on the displacement 80,. Hence, if the 
force acting at time t on an element of the string pdxhe p Ydx, 

^.^fpYsin^dx (8). 

In these equations 0« is a linear quantity, as we see from (1); and 
4>, is therefore a force of the ordinary kind. 

129. In the applications that we have to make, the only 
impressed force will be supposed to act in the immediate neigh- 
bourhood of one point x^b, and may usually be reckoned as 
a whole, so that 

4>, = sin -^ I /jFda? (1). 

If the point of application of the force coincide with a node of 
the mode («), 4>, = 0, and we learn that the force is altogether 
without influence on the component in question. This principle 
is of great importance ; it shews, for example, that if a string be 
at rest in its position of equilibrium, no force applied at its centre, 
whether in the form of plucking, striking, or bowing, can generate 
any of the even normal components ^ If after the operation of 
the force, its point of application be damped, as by touching it 

1 The observation that a harmonio is not generated, when one of its nodal 
points ia plucked, is due to Young. 



188 TRANSVERSE VIBRATIONS OF STRINGS. [l29. 

with the fioger, all motion must forthwith cease ; for those com- 
ponents which have not a node at the point in question are 
stopped by the damping, and those which have, are absent from 
the beginning^ More generally, by damping any point of a 
sounding string, we stop all the component vibrations which have 
not, and leave entirely unaflfected those which have a node at the 
point touched. 

The case of a string pulled aside at one point and afterwards 
let go from rest may be regarded as included in the preceding 
statements. The complete solution may be obtained thus. Let 
the motion commence at the time f = 0; from which moment 
4>, = 0. The value of 0, at time t is 

^* = (^*)oCosn^4--(</),)o8inn^ (2), 

where {(f>g)o, {4>»)o denote the initial values of the quantities 
aflFected with the suffix «. Now in the problem in hand {4>t)o = 0, 
and (0,)o is determined by 

„.(^.), = |<I.. = |Fsinf (3). 

if F' denote the force with which the string is held aside at the 
point b. Hence at time t 

^''^Tpn'' sin-y-cosn« (4), 

and by (1) of §128 

, 2 ^. ^ «.« . sirb . STTX cos nt .^. 

where n = sira/L 

The symmetry of the expression (5) in x and b is an example 
of the principle of § 107. 

The problem of determining the subsequent motion of a string 
set into vibration by an impulse acting at the poiut 6, may be 
treated in a similar manner. Integrating (6) of § 128 over the 
duration of the impulse, we find ultimately, with the same nota- 
tion as before, 

. . 2 . sirb ^ 

(<^*)o = ^sm -J- F„ 

^ A like restilt ensnes when the point which is damped is at the same distance 
from one end of the string as the point of excitation is from the other end. 



129.] INITIAL CONDITIONS. 189 

if lY'dt be denoted by Fj. At the same time (<^,)o = 0, so that by 

(2) at time t 

2Yi ^^m . 8vb . STTX Binnt .^. 

y=i^^->"°"T''°"r-^ ^^^- 

The series of component vibrations is less convergent for a struck 
than for a plucked string, as the preceding expressions shew. 
The reason is that in the latter case the initial value of y is 
continuous, and only dyjdx discontinuous, while in the former it 
is y itself that makes a sudden spring. See §§ 32, 101. 

The problem of a string set in motion by an impulse may also 

be solved by the general formulae (7) and (8) of § 128. The force 

finds the string at rest at ^ = 0, and acts for an infinitely short 

time from ^ = to t = T, Thus (<^,)o and (<^,)o vanish, and (7) 

of 5 128 reduces to 

2 f^' 

while by (8) of §128 

|''<I>,d^' = sin?^|V'd^' = sin?^-^ F,. 

H^nce, as before, 

<f>i = j~ Fisin— ^ sinw^ (7). 

Hitherto we have supposed the disturbing force to be concen- 
trated at a single point. If it be distributed over a distance /3 
on either side of 6, we have only to integrate the expressions (6) 
and (7) with respect to 6, substituting, for example, in (7) in place 
of Fi sin (sirb/l), 



I 



F/ sin -J- db. 



If F/ be constant between the limits, this reduces to 

Fi — sm —f- sm — y— (8). 

STT t 

The principal effect of the distribution of the force is to render 
the series for y more convergent. 

130. The problem which will next engage our attention is 
that of the pianoforte wire. The cause of the vibration is here 
the blow of a hammer, which is projected against the string, and 



190 TRANSVERSE VIBRATIONS OF Sl'RINGS. [l30. 

after the impact rebounds. But we should not be justified in 
assuming, as in the last section, that the mutual action occupies 
so short a time that its duration may be neglected. Measured by 
the standards of ordinary life the duration of the contact is indeed 
very small, but here the proper comparison is with the natural 
periods of the string. Now the hammers used to strike the wires 
of a pianoforte are covered with several layers of cloth for the 
express purpose of making them more yielding, with the eflFect of 
prolonging the contact. The rigorous treatment of the problem 
would be difficult, and the solution, when obtained, probably too 
complicated to be of use ; but by introducing a certain simplifica- 
tion Helmholtz has obtained a solution representing all the 
essential features of the case. He remarks that since the actual 
yielding of the string must be slight in comparison with that of 
the covering of the hammer, the law of the force called into play 
during the contact must be nearly the same as if the string were 
absolutely fixed, in which case the force would vary very nearly as 
a circular function. We shall therefore suppose that at the time 
^ = 0, when there are neither velocities nor displacements, a force 
F sin pt begins to act on the string at a? = 6, and continues through 
half a period of the circular function, that is, until t = 7r/p, after 
which the string is once more free. The magnitude of p ^vill 
depend on the mass and elasticity of the hammer, but not to any 
great extent on the velocity with which it strikes the string. 

The required solution is at once obtained by substituting for 
4>, in the general formula (7) of § 128 its value given by 

<I>, = ^ sm^sinpt' (1), 

the range of the integration being from to ir/p. We find 
(t > ir/p) 



2F . 87rb fp , .. .,. . ., ,w 
<l>t = j — sm -J- I smn^t — t) ainpt at 



Ifip I JO 

nir 



4p cos ^ , . , 

-^. ..rsm -7- .smwU-s- (2), 



lfm{p^-n^y I ' \ 2pj 

and the final solution for y becomes, if we substitute for n and /> 
their values, 



cos -^- , . sm 



ifapl'F ^^<» 2pl I . STTW . sira/. 7r\ .^. 

y — ^JL 2 r-- . „ ,v sm— , sm-^— U-^p-)...(d). 



130.] PIANOFORTE STRING. 191 

We see that all components vanish which have a node at the 
point of excitement, but this conclusion does not depend on any 
particular law of force. The interest of the present solution lies 
in the information that may be elicited from it as to the depend- 
ence of the resulting vibrations on the duration of contact. If 
we denote the ratio of this quantity to the fundamental period of 
the string by v, so that 1/ = Tra : 2p/, the expression for the ampli- 
tude of the component 8 is 

^Fl V cos {sTTv) . sirh . . 

7i^-5(l-4^i;«)^"'"r ^*^- 

We fall back on the case of an impulse by putting i/ = 0, 



and 



F,= Fsinptdt^ — . 
Jo ^ p 



When V is finite, those components disappear, whose periods 
are |, ^, f, ... of the duration of contaxjt; and when 8 is very 
great, the series converges with 5""'. Some allowance must also 
be made for the finite breadth of the hammer, the effect of which 
will also be to favour the convergence of the series. 

The laws of the vibration of strings may be verified, at least 
in their main features, by optical methods of observation— either 
with the vibration-microscope, or by a tracing point recording the 
character of the vibration on a revolving drum. This character 
depends on two things, — the mode of excitement, and the point 
whose motion is selected for observation. Those components do 
not appear which have nodes either at the point of excitement, or 
at the point of observation. The former are not generated, and 
the latter do not manifest themselves. Thus the simplest motion 
is obtained by plucking the string at the centre, and observing 
one of the points of trisection, or vice versa. In this case the 
first harmonic which contaminates the purity of the principal 
vibration is the fifth coYnponent, whose intensity is usually in- 
sufficient to produce much disturbance. 

[The dynamical theory of the vibration of strings may be 
employed to test the laws of hearing, and the necessary experi- 
ments are easily carried out upon a grand pianoforte. Having 
freed a string, say c, from its damper by pressing the digital, pluck 
it at one-third of its length. According to Young's theorem the 
third component vibration is not excited then, and in corre- 



192 TRANSVERSE VIBRATIONS OF STRINGS. [l30. 

spoDdence with that fact the ear fails to detect the component g\ 
A slight displacement of the point plucked brings g' in again; 
and if a resonator {g[) be used to assist the ear, it is only with 
difficulty that the point can be hit with such precision as entirely 
to extinguish the tone. Experiments of this kind shew that the 
ear analyses the sound of a string into precisely the same con- 
stituents as are found by sympathetic resonance, that is, into 
simple tones, according to Ohm's definition of this conception. 
Such experiments are also well adapted to shew that it is not a 
mere play of imagination when we hear overtones, as some people 
believe it is on hearing them for the first time\ 

If, after the string has been sounded loudly by striking the 
digital, it be touched with the finger at one of the points of 
trisection, all components are stopped except the 3rd, 6th, &c., so 
that these are left isolated. The inexperienced observer is usually 
surprised by the loudness of the residual sound, and begins to 
appreciate the large part played by overtones.] 

131. The case of a periodic force is included in the general 
solution of § 128, but we prefer to follow a somewhat different 
method, in order to make an extension in another direction. We 
have hitherto taken no account of dissipative forces, but we will 
now suppose that the motion of each element of the string is 
resisted by a force proportional to its velocity. The partial 
differential equation becomes 

by means of which the subject may be treated. But it is still 
simpler to avail ourselves of the results of the last chapter, 
remarking that in the present case the dissipation-function F is 
of the same form as T. In fact 



•» . 



^=ip^/.2^,<^.» (2), 

where ^i, ^j,... are the normal co-ordinates, by means of which 
T and V are reduced to sums of squares. The equations of 

motion are therefore simply 

2 
^, + /t<^, + n«</>.= r^4>, (3), 

1 Helmholtz, Ch. n, ; Brandt, Pogg, Arm,, Vol. cni. p. 824, 1861. 



131.] FRICTION PROPORTIONAL TO VELOCITY. 193 

of the same form as obtains for systems with but one degree of 
freedom. It is only necessary to add to what was said in 
Chapter iii., that since k is independent of 8y the natural vibra- 
tions subside in such a manner that the amplitudes maintain their 
relative values. 

If a periodic force F cos pt act at a single point, we have 

<I>, = -F8in -, cospt (4), 

and§46 <^,= lppK~^^^ ^- cos(pt-€) (o), 

where tan 6= ^ — ~ (6). 

If among the natural vibrations there be any one nearly 
isochronous with cosp^, then a large vibration of that type will 
be forced, unless indeed the point of excitement should happen to 
fall near a node. In the case of exact coincidence, the component 
vibration in question vanishes ; for no force applied at a node can 
generate it, under the present law of friction, which however, it 
may be remarked, is very special in character. If there be no 
friction, /c = 0, and 

^p<^f = ^i-^8in-^- cospf (/), 

which would make the vibration infinite, in the case of perfect 
isochronism, unless sin (sTrb/l) = 0. 

The value of y is here, as usual, 

y = <^sm-^ -f </»jSm y- + ^sm -7-+ (8). 

132. The preceding solution is an example of the use of 
normal co-ordinates in a problem of forced vibrations. It is of 
course to free vibrations that they are more especially applicable, 
and they may generally be used with advantage throughout, 
whenever the system after the operation of various forces is 
ultimately left to itself. Of this application we have already had 
examples. 

In the case of vibrations due to periodic forces, one advantage 
of the use of normal co-ordinates is the facility of comparison with 
the equilibrium theory, which it will be remembered is the theory 

R. 13 



194 TRANSVERSE VIBRATIONS OF STRINGS. [l32. 

of the motion on the supposition that the inertia of the system 
may be left out of account. If the value of the normal co-ordinate 
<f>t on the equilibrium theory be A g cos pt, then the actual value 
will be given by the equation 

^• = ri^^^^^l^^ W' 

so that, when the result of the equilibrium theory is known and 
can readily be expressed in terms of the normal co-ordinates, the 
true solution with the eflfects of inertia included can at once be 
written down. 

In the present instance, if a force F cos pt of very long period 
act at the point b of the string, the result of the equilibrium 
theory, in accordance with which the string would at any moment 
consist of two straight portions, will be 

Ip^t^^sm ^ cospt (2), 

from which the actual result for all values of p is derived by simply 
writing (n' — p^) in place of n\ 

The value of y in this and similar cases may however be 
expressed in finite terms, and the difficulty of obtaining the 
finite expression is usually no greater than that of finding the 
form of the normal functions when the system is free. Thus in 
the equation of motion 

suppose that F varies as cos moL The forced vibration will then 
satisfy 

S+-'^ = -i»^ <3)- 

If F= 0, the investigation of the normal functions requires the 
solution of 

and a subsequent determination of m to suit the boundary con- 
ditions. In the problem of forced vibrations m is given, and we 
have only to supplement any particular solution of (3) with the 
complementary function containing two arbitrary constants. This 
function, apart from the value of m and the ratio of the constants. 



132.] COMPARISON WITH EQUILIBRIUM THEORY. 195 

is of the same form as the normal functions ; and all that remains to 
be effected is the determination of the two constants in accordance 
with the prescribed boundary conditions which the complete 
solution must satisfy. Similar considerations apply in the case 
of any continuous system. 

133. If a periodic force be applied at a single point, there are 
two distinct problems to be considered; the first, when at the 
point x^by a, given periodic force acts ; the second, when it is the 
actual motion of the point b that is obligatory. But it will be 
convenient to treat them together. 

The usual differential equation 

is satisfied over both the parts into which the string is divided at 
6, but is violated in crossing from one to the other. 

In order to allow for a change in the arbitrary constants, we 
must therefore assume distinct expressions for y, and afterwards 
introduce the two conditions which must be satisfied at the point 
of junction. These are 

(1) That there is no discontinuous change in the value of y ; 

(2) That the resultant of the tensions acting at b balances the 
impressed force. 

Thus, \i Foo^pt be the force, the second condition gives 



2'.A@+i'cos/)« = (2), 



where A{dy/dx) denotes the alteration in the value of dy/dx 
incurred in crossing the point a? = 6 in the positive direction. 

We shall, however, find it advantageous to replace cospt by 
the complex exponential e^^^, and finally discard the imaginary 
part, when the symbolical solution is completed. On the assump- 
tion that y varies as e'^^ the differential equation becomes 

S + '^V'O (3); 

where X^ is the complex constant, 

^' = ^(P'-»1>«) (4). 

13—2 



196 TRANSVERSE VIBRATIONS OF STRINGS. [l33. 

The most general solution of (3) consists of two terms, pro- 
portional respectively to sinXo;, and cosXo;; but the condition to 
be satisfied at a? = shews that the second does not occur here. 
Hence if 7 ff^ be the value of y at a? = 6, 

sinXa? .^ ,,. 

y = 7-^^ri-«'* (5), 

is the solution applying to the first part of the string from a; == 

to a; = 6. In like manner it is evident that for the second part we 

shall have 

sinX(Z — a?)^^ 

If 7 be given, these equations constitute the sjrmbolical solution 
of the problein ; but if it be the force that is given, we require 
further to know the relation between it and 7. 

Differentiation of (5) and (6) and substitution in the equation 
analogous to (2) gives 



Thus 



_ F_ sinXA sinX(Z--6) 
"^"T, XsinXi ^^)- 



^F sm\x sin \{l — 6) ,^ 
^"2\ Xsinx; ^ 

from 0? = to d; = 6 
__ ^ sin X(Z — x) sin X6 ^^ 

from a? = 6 to a? = Z 



L..(8y. 



These equations exemplify the general law of reciprocity 
proved in the last chapter ; for it appears that the motion at x 
due to the force at h is the same as would have been found at h, 
had the force acted at x. 

In discussing the solution we will take first the case in which 
there is no friction. The coefficient k is then ^ zero; while X is 
real, and equal to pjcu The real part of the solution, correspond- 
ing to the force Foospt, is found by simply putting cos|>^ for ^^^ 
in (8), but it seems scarcely necessary to write the equations again 
for the sake of so small a change. The same remark applies to 
the forced motion given in terms of 7. 

It appears that the motion becomes infinite in case the force 

^ Donkin's Acouttici, p. 121. 



133.] PKMODIC FORCE AT ONE POINT. 197 

is isochronous with one of the natural vibrations of the entire 
string, unless the point of application be a node ; but in practice 
it is not easy to arrange that a string shall be subject to a force 
of given magnitude. Perhaps the best method would be to attach 
a small mass of iron, attracted periodically by an electro-magnet, 
whose coils are traversed by an intermittent current. But unless 
some means of compensation were devised, the mass would have 
to be very small in order to avoid its inertia introducing a new 
complication. 

A better approximation may be obtained to the imposition of 
an obligatory motion. A massive fork of low pitch, excited by 
a bow or sustained in permanent operation by electro-magnetism, 
executes its vibrations in approximate independence of the re- 
actions of any light bodies which may be connected with it. In 
order therefore to subject any point of a string to an obligatory 
transverse motion, it is only necessary to attach it to the extremity 
of one prong of such a fork, whose plane of vibration is perpendicular 
to the length of the string. This method of exhibiting the forced 
vibrations of a string appears to have been first used by Melded 

Another arrangement, better adapted for aural observation, 
has been employed by Helmholtz. The end of the stalk of a 
powerful tuning-fork, set into vibration with a bow, or otherwise, 
is pressed against the string. It is advisable to file the surface, 
which comes into contact with the string, into a suitable (saddle- 
shaped) form, the better to prevent slipping and jarring. 

Referring to (5) we see that, if sin X& vanished, the motion 
(according to this equation) would become infinite, which may be 
taken to prove that in the case contemplated, the motion would 
really become great, — so great that corrections, previously insigni- 
ficant, rise into importance. Now sin X6 vanishes, when the force 
is isochronous with one of the natural vibrations of the first part 
of the string, supposed to be held fixed at and b. 

When a fork is placed on the string of a monochord, or other 
instrument properly provided with a sound-board, it is easy to 
find by trial the places of maximum resonance. A very slight 
displacement on either side entails a considerable falling off in the 
volume of the sound. The points thus determined divide the 
string into a number of equal parts, of such length that the 
natural note of any one of them (when fixed at both ends) is 

1 Pogg. Ann, oix. p. 198, 1859. 



198 TRANSVERSE VIBRATIONS OF STRINGS. [l33. 

the same as the note of the fork, as may readily be verified. The 
important applications of resonance which Helmholtz has made to 
purify a simple tone from extraneous accompaniment will occupy 
our attention later. 

134. Returning now to the general case where X is complex, 
we have to extract the real parts from (5), (6), (8) of § 133. For 
this purpose the sines which occur as factors, must be reduced to 
the form Re^. Thus let 

sin\a? = i2a;e*** (1), 

with a like notation for the others. From (5) § 133 we shall thus 
obtain 

from a; = to a? = 6, 
and from (6) § 133 

from a; = 6 to 0? = Z, 

corresponding to the obligatory motion y = 7 cosp^ at 6. 

By a similar process from (8) § 138, if 

X = a+i/8 (3), 

we should obtain 



y = 7;g-cos(pe + €a.-66) (2), 



J'=lv(fc^.''^(^^'^'*^'""''"'^''"^^/"0 



from a? = to a? = 6 



,..(4), 



from a? = 6toa? = Z j 

corresponding to the impressed force F^o^pi at 6. It remains to 
obtain the forms of Rx^^xj &c. 

The values of a and y8 are determined by 

a3-^ = £!. 2a^ = -^ (5), 

and sin Xo; = sin our cos i^x + cos fxx sin ifix 

= sm fix s f- * cos ow? s , 



134. J FRICTION PROPORTIONAL TO VELOCITY. 199 

SO that 

Bx^ = 8in^ax ( ^ — j H-cos^ow? f 5 j ...(6), 



while 



V(a» + /8') = ^^(/)*+pV) (8). 



This completes the solution. 

If the friction be very small, the expressions may be simpli- 
fied. For instance, in this case, to a sufficient approximation, 

a -^pja, ;8 = - /c/2a, V(a' + P") ^Vl^^ 
i(e^* + e-^*) = l, i (e^* - e-^*) = - /ca;/2a ; 

so that, corresponding to the obligatory motion at 6 y = 7 cosp^, the 
amplitude of the motion between a? = and a; = 6 is, approximately 



/ . ^px i^a? ^px \ 
^m-^- + -^ - cos'-^ I 



71 



. 'p6 H^h^ ^ph 
sm'^— + -/- cos*^— 
a 4a* a j 



(9), 



which becomes great, but not infinite, when sin (pb/a) = 0, or the 
point of application is a node. 

If the imposed force, or motion, be not expressed by a single 
harmonic tenn, it must first be resolved into such. The preceding 
solution may then be applied to each component separately, and 
the results added together. The extension to the case of more than 
one point of application of the impressed forces is also obvioua 
To obtain the most general solution satisfying the conditions, the 
expression for the natural vibrations must also be added; but 
these become reduced to insignificance after the motion has been 
in progress for a sufficient time. 

The law of friction assumed in the preceding investigation is 
the only one whose results can be easily followed deductively, and 
it is sufficient to give a general idea of the effects of dissipative 
forces on the motion of a string. But in other respects the con- 
clusions drawn from it possess a fictitious simplicity, depending on 
the fact that F — the dissipation-function — is similar in form to T, 
which makes the normal co-ordinates independent of each other. 



200 TRANSVEKSE VIBRATIONS OP STRINGS. [134. 

In almost any other case (for example, when but a single point of 
the string is retarded by friction) there are no normal co-ordinates 
properly so called. There exist indeed elementary types of vibra- 
tion into which the motion may be resolved, and which are 
perfectly independent, but these are essentially diflferent in cha- 
racter from those with which we have been concerned hitherto, for 
the various parts of the system (as affected by one elementary 
vibration) are not simultaneously in the same phase. Special cases 
excepted, no linear transformation of the co-ordinates (with real 
coefficients) can reduce T, F, and V together to a sum of 
squares. 

If we suppose that the string has no inertia, so that T = 0, 
F and V may then be reduced to sums of squares. This problem 
is of no acoustical importance, but it is interesting as being 
mathematically analogous to that of the conduction and radiation 
of heat in a bar whose ends are maintained at a constant tem- 
perature. 

136. Thus far we have supposed that at two fixed points, 
07 = and x^l, the string is held at rest. Since absolute fixity 
cannot be attained in practice, it is not without interest to inquire 
in what manner the vibrations of a string are liable to be modified 
by a yielding of the points of attachment; and the problem 
will furnish occasion for one or two remarks of importance. 
For the sake of simplicity we shall suppose that the system is 
sjrmmetrical with reference to the centre of the string, and that 
each extremity is attached to a mass M (treated as unextended in 
space), and is urged by a spring (ji) towards the position of equi- 
librium. If no frictional forces act, the motion is necessarily 
resolvable into normal vibrations. Assume 

y = {a sin 7IM7 + y8 cos ma?} cos(?na^— €) (1). 

The conditions at the ends are that 

when a? = 0, My + fiy-^ T,^ 

^ \ (2), 

when x^ly My + /xy = — Ti -i^ 

which give 

(!t__^ tan mZ — a _ /i — Ma^m^ .^. 

i8~atanwiZ + /8" ~mTi ^ ^' 



135.] EXTREMITIES SUBJECT TO YIELDING. 201 

two equations, sufficient to determine m, and the ratio of /9 to a. 
Eliminating the latter ratio, we find 

taLnml = - (4), 

if for brevity we write v for ^j . 

Equation (3) has an infinite number of roots, which may be 
found by writing tan for v, so that tan mZ = tan 20, and the result 
of adding together all the corresponding particular solutions, each 
with its two arbitrary constants a and 6, is necessarily the most 
general solution of which the problem is capable, and is therefore 
adequate to represent the motion due to an arbitrary initial dis- 
tribution of displacement and velocity. We infer that any function 
of X may be expanded between a? = and x = liu a series of terms 

(fhi^i sin vi^x + cos niix) + <f>i{p2 sin rryc + cos rrifX) + (5), 

7^1, rwa, &c. being the roots of (3) and Vi, Vj, &c. the corresponding 
values of p. The quantities ^i, ^3, &c. are the normal co-ordinates 
of the system. 

From the s)rmmetry of the system it follows that in each 
normal vibration the value of y is numerically the same at points 
equally distant from the middle of the string, for example, at the 
two ends, where x = and x = l. Hence 1;, sin m^ + cos m^ = ± 1 , 
as may be proved also from (4). 

The kinetic energy T of the whole motion is made up of the 
energy of the string, and that of the masses M. Thus 

^~i/^| {S ^ (1/ sin TTW? -h cos ma?)}^ da? 
J 

+ )^M {<^i + <^j H- . . .}' 4- iif (<^i (i/i sin mil + cos mj,) -[-...}'. 

But by the characteristic property of normal co-ordinates, terms 
containing their products cannot be really present in the expres- 
sion for T, so that 

p 1 (Pr sin mfX H- cos mrx) (i/, sin m^ + cos m^x) dx 

+ if H- M(pr sin mfl H- cos rrirl) (v, sin m^ 4- cos m^ = (6), 

if r and 8 be different. 

This theorem suggests how to determine the arbitrary con- 



202 TRANSVERSE VIBRATIONS OF STRINGS. [l35. 

stants, 80 that the series (5) may represent an arbitrary function 
y. Take the expression 

pj y(y» 81^ ^r!P + cos mgx)dx + My^ + Myi {v, sin mJL + cos mJL). . .(7), 

and substitute in it the series (5) expressing y. The result is a 
series of terms of the type 

p\ ^r (vr sin mra? + cos m^a?) (v, sin m^ + cos m,aj) da? 

J 

H- M^ H- M<l>r {vr sin mji + cos mji) {vg sin m^ + cos ntgl), 

all of which vanish by (6), except the one for which r = «. Hence 
(f>g is equal to the expression (7) divided by 

p I (vg sin rrigX + cos mgxy dx + M+M^i/g sin mj + cos m^y. , .(8), 

J 

and thus the coefficients of the series are determined. If M=0, 
even although /i be finite, the process is of course much simpler, 
but the unrestricted problem is instructive. So much stress is 
often laid on special proofs of Fourier's and Laplace's series, that 
the student is apt to acquire too contracted a view of the nature 
of those important results of analysis. 

We shall now shew how Fourier's theorem in its general form 
can be deduced from our present investigation. Let ilf = ; then 
if /i = X , the ends of the string are fast, and the equation de- 
termining m becomes tan mi = 0, or ml = sw, as we know it must 
be. In this case the series for y becomes 

y — AiSm^ + A^sin —j- H-il,sm— p + (9), 

which must be general enough to represent any arbitrary functions 
of Xy vanishing at and 2, between those limits. But now suppose 
that /i is zero, M still vanishing. The ends of the string may be 
supposed capable of sliding on two smooth rails perpendicular to 
its length, and the terminal condition is the vanishing of dyjdx. 
The equation in m is the same as before ; and we learn that any 
function y whose rates of variation vanish at a? = and x = l, can 
be expanded in a series 

y = jBi cos -V- H- jBj cos -,- H-^,cos— r- + (10). 



135.] Fourier's theorem. 203 

This series remains unaifected when the sign of a; is changed, 
and the first series merely changes sign without altering its 
numerical magnitude. If therefore yf be an even function of x, 
(10) represents it from — Z to H- f. And in the same way, if y be 
an odd function of x, (9) represents it between the same limita 

Now, whatever function of x <f> (x) may be, it can be divided 
into two parts, one of which is even, and the other odd, thus : 

A (^) = * 1?)_± ^(-^) + <^ (a?) - <^ (- a?) . 

so that, if <f> (x) be such that ^ (- Z) = <^ (+ 1) and 4>' (- 1) = <(> (+ 1), 
it can be represented between the limits ±lhy the mixed series 

. . Tra? . o TTX . , 2irx „ 2Trx „-. 

-4iSin-^ +jDiC0S-^+-4jSm , -h -B, cos -t— + (11). 

This series is periodic, >vith the period 21. If therefore ^ {x) 
possess the same property, no matter what in other respects its 
character may be, the series is its complete equivalent. This is 
Fourier's theorem \ 

We now proceed to examine the effects of a slight yielding of 
the supports, in the case of a string whose ends are approximately 
fixed. The quantity v may be great, either through fi or through 
M. We shall confine ourselves to the two principal cases, (1) 
when /i is great and M vanishes, (2) when /x vanishes and M is 
great. 

In the first case v = ^(— , 

lim 

and the equation in m is approximately 

, 2 2Tim 
tan mt = — = . 

V fl 

Assume mZ = ^tt + x, where x is small ; then 

2T,.sir ... 

X = tan x = -, — approximately, 

and mi = 57r(l--^') (12). 

^ The best * syetem ' for proving Fourier's theorem from dynamical considera- 
tions is an endless chain stretched round a smooth cylinder (§ 189), or a thin 
re-entrant column of air enclosed in a ring-shaped tube. 



204 TRANSVERSE VIBRATIONS OF STRINGS. [135. 

To this order of approximation the tones do not cease to form 
a harmonic scale, but the pitch of the whole is slightly lowered. 
The effect of the yielding is in &ct the same as that of an increase 

2T 

in the length of the string in the ratio 1 : 1 H — ^ , as might 

have been anticipated. 

The result is otherwise if fi vanish, while M is great. Here 

and tan ml = ^ ^ approximately. 

Hence mZ = g^r + i# , ^ — (13)- 

Ma^ . sir 

The effect is thus equivalent to a decrease in / in the ratio 

1.1- 1?^ 

and consequently there is a rise in pitch, the rise being the 
greater the lower the component tone. It might be thought 
that any kind of jdelding would depress the pitch of the string, 
but the preceding investigation shews that this is not the case. 
Whether the pitch will be raised or lowered, depends on the 
sign of i;, and this again depends on whether the natural note of 
the mass M urged by the spring /a is lower or higher than that of 
the component vibration in question. 

136. The problem of an otherwise uniform string carrying 
a finite load Jf at a? = 6 can be solved by the formulae investigated 
in § 133. For, if the force F cos pt be due to the reaction against 
acceleration of the mass M, 

F^ypW. (1), 

which combined with equation (7) of § 133 gives, to determine the 
possible values of X (or jp : a), 

a»ilfXsinX6 sinX. (i - 6) = T^sinXZ (2). 

The value of y for any normal vibration corresponding to X is 

y = P sin Xa; sin X (Z — 6) cos (aXt — e) 

from x=^0 to x = b . .^. 

y = PsinX (? — a:) sin X6 cos (aX< — c) ' 

from x=^b to x = l 

where P and € are arbitrary constants. 



136.] FINITE LOAD. 205 

It does not require analysis to prove that any normal com- 
ponents which have a node at the point of attachment are un- 
affected by the presence of rthe load. For instance, if a string be 
weighted at the centre, its component vibrations of even orders 
remain unchanged, while all the odd components are depressed in 
pitch. Advantage may sometimes be taken of this effect of a 
load, when it is desired for any purpose to disturb the harmonic 
relation of the component tones. 

If M be very great, the gravest component is widely sepa- 
rated in pitch from all the others. We will take the case when 
the load is at the centre, so that 6 = 2 — 6 = ^2. The equation in 
\ then becomes 

sm 2" . j-g tan 2- - y = (4), 

where pi : M, denoting the ratio of the masses of the string and 
the load, is a small quantity which may be called a?. The first 
root corresponding to the tone of lowest pitch occurs when \\l ia 
small, and such that 

(JXZ)» {1 4 i (i ny] = o' nearly, 

whence i[Kl = a (1 — ^ o*), 

and the periodic time is given by 



/Ml 



['^t^ (^)- 



The second term constitutes a correction to the rough value 
obtained in a previous chapter (§ 52), by neglecting the inertia of 
the string altogether. That it would be additive might have 
been expected, and indeed the formula as it stands may be ob- 
tained from the consideration that in the actual vibration the two 
parts of the string are nearly straight, and may be assumed to be 
exactly so in computing the kinetic and potential energies, with- 
out entailing any appreciable error in the calculated period. On 
this supposition the retention of the inertia of the string increases 
the kinetic energy corresponding to a given velocity of the load in 
the ratio of if : Jf+Jpi, which leads to the above result. This 
method has indeed the advantage in one respect, as it might be 
applied when p is not uniform, or nearly uniform. All that is 
necessary is that the load M should be sufSciently predominant. 

There is no other root of (4), until sinJXZ = 0, which gives 



M» W ^^^i*^ 



me 



206 



TRANSVERSE VIBRATIONS OF STRINGS. 



[136. 



the second component of the string, — a vibration independent of 
the load. The roots after the first occur in closely contiguous 
pairs; for one set is given by J\Z = 57r, and the other approxi- 
mately by iXZ=57r4--Ar>, in which the second term is small. 
The two types of vibration for « = 1 are shewn in the figure. 

Fig. 21. 





The general formula (2) may also be applied to find the effect 
of a small load on the pitch of the various components. 

137. Actual strings and wires are not perfectly flexible. 
They oppose a certain resistance to bending, which may be divided 
into two parts, producing two distinct effects. The first is called 
viscosity, and shews itself by damping the vibrations. This part 
produces no sensible effect on the periods. The second is con- 
servative in its character, and contributes to the potential energy 
of the system, with the effect of shortening the periods. A com- 
plete investigation cannot conveniently be given here, but the 
case which is most interesting in its application to musical 
instruments, admits of a sufficiently simple treatment. 

When rigidity is taken into account, something more must be 
specified with respect to the terminal conditions than that y 
vanishes. Two cases may be particularly noted : — 

(i) When the ends are clamped, so that dy/dx = at the ends. 



137.] CORRECTION FOR RIGIDITY. 207 

(ii) When the terminal directions are perfectly free, in which 
case dhf/da^ = 0. 

It is the latter which we propose now to consider. 

If there were no rigidity, the type of vibration would be 

STTtXi 

y X sin . , satisfying the second condition. 

The eflFect of the rigidity might be slightly to disturb the type; 
but whether such a result occur or not, the period calculated 
from the potential and kinetic energies on the supposition that 
the type remains unaltered is necessarily correct as far as the first 
order of small quantities (§ 88). 

Now the potential energy due to the stiffness is expressed by 



SF 



=**/:(2)'^ «• 



where £ is a quantity depending on the nature of the material 
and on the form of the section in a manner that we are not now 
prepared to examine. The /o7^n of SF is evident, because the force 
requiied to bend any element ds is proportional to ds, and to the 
amount of bending already effected, that is to ds/p. The whole 
work which must be done to produce a curvature l/p in ds is 
therefore proportional to ds/p^; while to the approximation to 
which we work d8 = dx, and l/p = d^yldx\ 



STTX 



Thus, if y = <^ sin — ^ 

and the period of <^ is given by 

if To denote what the period would become if the string were 
endowed with perfect flexibility. It appears that the effect of the 
stiffness increases rapidly with the order of the component vibra- 
tions, which cease to belong to a harmonic scale. However, in the 
strings employed in music, the tension is usually sufficient to 
reduce the influence of rigidity to insignificance. 

The method of this section cannot be applied without modifi- 
cation to the other case of terminal condition, namely, when the 
ends are clamped. In their immediate neighbourhood the type of 



208 TRANSVERSE VIBRATIONS OF STRINGS. [l37. 

vibration must differ from that assumed by a perfectly flexible 
string by a quantity, which is no longer small, and whose square 
therefore cannot be neglected. We shall return to this subject, 
when treating of the transverse vibrations of rods. 

138. There is one problem relating to the vibrations of strings 
which we have not yet considered, but which is of some practical 
interest, namely, the character of the motion of a violin (or cello) 
string under the action of the bow. In this problem the modus 
operandi of the bow is not sufficiently understood to allow us to 
follow exclusively the a priori method : the indications of theory 
must be supplemented by special observation. By a dexterous 
combination of evidence drawn from both sources Helmholtz has 
succeeded in determining the principal features of the case, but 
some of the details are still obscure. 

Since the note of a good instrument, well handled, is musical, 
we infer that the vibrations are strictly periodic, or at least that 
strict periodicity is the ideal. Moreover — and this is very import- 
ant — the note elicited by the bow has nearly, or quite, the same 
pitch as the natural note of the string. The vibrations, although 
forced, are thus in some sense free. They are wholly dependent 
for their maintenance on the energy drawn fi-om the bow, and yet 
the bow does not determine, or even sensibly modify, their periods. 
We are reminded of the self-acting electrical interrupter, whose 
motion is indeed forced in the technical sense, but has that kind 
of freedom which consists in determining (wholly, or in part) under 
what influences it shall come. 

But it does not at once follow from the fact that the string 
vibrates with its natural periods, that it conforms to its natural 
types. If the coefficients of the Fourier expansion 

. . irx . . 2,irx 
y=^<l>i&m -y- + <^, sm -y- + 



be taken as the independent co-ordinates by which the configura- 
tion of the system is at any moment defined, we know that when 
there is no friction, or friction such that Foe T, the natural vibra- 
tions are expressed by making each co-ordinate a simple harmonic 
(or quasi-harmonic) function of the time ; while, for all that has 
hitherto appeared to the contrary, each, co-ordinate in the present 
case might be any function of the time periodic in time t. But a 



138.] VIOLIN STRING. 209 

little examination will shew that the vibrations must be sensibly 
natural in their types as well as in their periods. 

The force exercised by the bow at its point of application may 
be expressed by 

F= ^Ar COS f €r] ', 

SO that the equation of motion for the co-ordinate (f>t is 

h being the point of application. Each of the component paits of 
4>, will give a corresponding term of its own period in the solu- 
tion, but the one whose period is the same as the natural period 
of <\>g will rise enormously in relative importance. Practically then, 
if the damping be small, we need only retain that part of ^« 

which depends on Ag cos ( e,] , that is to say, we may regard 

the vibrations as natural in their types. 

Another material fact, supported by evidence drawn both from 
theory and aural observation, is this. All component vibrations 
are absent which have a node at the point of excitation. "In 
order, however, to extinguish these tones, it is necessary that the 
coincidence of the point of application of the bow with the node 
should be very exact. A very small deviation reproduces the 
missing tones with considerable strengths" 

The remainder of the evidence on which Helmholtz' theory 
rests, was derived from direct observation with the vibration- 
microscope. As explained in Chapter ii., this instrument affords 
a view of the curve representing the motion of the point under 
observation, as it would be seen traced on the surface of a trans- 
parent cylinder. In order to deduce the representative curve in 
its ordinary form, the imaginary cylinder must be conceived to 
be unrolled, or developed, into a plane. 

The simplest results are obtained when the bow is applied at a 
node of one of the higher components, and the point observed is 
one of the other nodes of the same system. If the bow work 
fairly so as to draw out the fundamental tone clearly and strongly, 
the representative curve is that shewn in figure 22; where the 

^ Donkin's Acousiict^ p. 131. 
R. 14 



210 



TRANSVERSE VIBRATIONS OF STRINGS. 



[138. 



abscissse correspond to the time (AB being a complete period), 
and the ordinates represent the displacement. The remarkable 

Fig. 22. 




fact is disclosed that the whole period t may be divided into two 
parts To and t — Tq, during each of which the velocity of the 
observed point is constant ; but the velocities to and fro are in 
general unequal. 

We have now to represent this curve by a series of harmonic 
terms. If the origin of time correspond to the point A, and 
AD^ FC = 7, Fourier's theorem gives 

y=» -r-r \2 ^jsm — ?sm — («-^) (1). 

^ 7r»To(T-To) •-! «* T T V 2/ ^ ^ 

With respect to the value of To, we know that all those com- 
ponents of y must vanish for which sin (sTrXo/l) = (x^ being the 
point of observation), because under the circumstances of the case 
the bow cannot generate them. There is therefore i*eason to 
suppose that TqI t=:Xq: l\ and in &ct observation proves that 
AC : CB (in the figure) is equal to the ratio of the two parts into 
which the string is divided by the point of observation. 

Now the free vibrations of the string are represented in 
general by 

STTX 

iAgCoa - 



f~ao 



y = 2,-i sm-j- j^,cos — +BgBm-yV; 



and this at the point x = Xo must agree with (1). For convenience 
of comparison, we may write 



. 2«7r^ . „ . 287rt ^ 28Tr 
Ag cos h Bg sm = Cg cos — 

T T T 



n • 257r 
+ Jjg sm 



and it then appears that Cg = 0. 



('-?) 

('-!•)• 



138.] VIOLIN STRING. 211 

We find also to determine D, 

. «7ra?o rv 27T* 1 . airXQ 

I tt'to (t — To) «* C 

whence 

unless sin (sirxo/l) = 0. 

In the case reserved, the comparison leaves Dg undetermined, 
but we know on other grounds that D, then vanishes. However, 
for the sake of simplicity, we shall suppose for the present that 
Df is always given by (2). If the point of application of the bow 
do not coincide with a node of any of the lower components, the 
error committed will be of no great consequence. 

On this understanding the complete solution of the problem is 

y=-r — J- . T -sin —J- sm U— 77 (3). 

^ 7r»To (t - To) ^-1 5* I T \ 2j ^^ 

The amplitudes of the components are therefore proportional to «~*. 
In the case of a plucked string we found for the corresponding 
function «~' sin (airb/l), which is somewhat similar. If the string 
be plucked at the middle, the even components vanish, but the 
odd ones follow the same law as obtains for a violin string. The 
equation (3) indicates that the string is alwa}rs in the form of two 
straight lines meeting at an angle. In order more conveniently 
to shew this, let us change the origin of the time, and the constant 
multiplier, so that 

y = — S^ain-psin^j^ W 

will be the equation expressing the form of the string at any time. 

Now we know (§ 127) that the equation of the pair of lines 
proceeding from the fixed ends of the string, and meeting at a 
point whose co-ordinates are a, 13, is 

2BP -» 1 . 8Tra . svx 

Thus at the time t, (4) represents such a pair of lines, meeting at 
the point whose co-ordinates are given by 

I3fi 



a(Z-a) 



= ±4P, 



sm -J- = ± sm - — 

I T 



14—2 



212 TRANSVERSE VIBRATIONS OF STRINGS. [l38. 

These equations indicate that the projection on the axis of x 
of the point of intersection moves uniformly backwards and 
forwards between a: = and a? = ?, and that the point of inter- 
section itself is situated on one or other of two parabolic arcs, 
of which the equilibrium position of the string is a common 
chord. 

Since the motion of the string as thus defined by that of the 
point of intersection of its two straight parts, has no especial 
relation to Xq (the point of observation), it follows that, according 
to these equations, the same kind of motion might be observed at 
any other point. And this is approximately true. But the theo- 
retical result, it will be remembered, was only obtained by as- 
suming the presence in certain proportions of component vibrations 
having nodes at x^^ though in fact their absence is required by 
mechanical laws. The presence or absence of these components is 
a matter of indifference when a node is the point of observation, 
but not in any other case. When the node is departed from, the 
vibration curve shews a series of ripples, due to the absence of 
the components in question. Some further details will be found 
in Helmholtz and Donkin. 

The sustaining power of the bow depends upon the fact that 
solid friction is less at moderate than at small velocities, so that 
when the part of the string acted upon is moving with the bow 
(not improbably at the same velocity), the mutual action is greater 
than when the string is moving in the opposite direction with 
a greater relative velocity. The accelerating effect in the first 
part of the motion is thus not entirely neutralised by the sub- 
sequent retardation, and an outstanding acceleration remains 
capable of maintaining the vibration in spite of other losses of 
energy. A curious effect of the same peculiarity of solid friction 
has been observed by W. Froude, who found that the vibrations 
of a pendulum swinging from a shaft might be maintained or 
even increased by causing the shaft to rotate. 

[Another case in which the vibrations of a string are main- 
tained is that of the Aeolian Harp. It has often been suggested 
that the action of the wind is analogous to that of a bow ; but the 
analogy is disproved by the observation* that the vibrations are 
executed in a plane transverse to the direction of the wind. The 
true explanation involves hydrodynamical theory not yet de- 
veloped.] 

1 Phil Mag., March, 1879, p. 161. 



139.] STRINGS STRETCHED ON CUR\nBD SURFACES. 213 

139. A string stretched on a smooth curved surface will in 
equilibrium lie along a geodesic line, and, subject to certain con- 
ditions of stability, will vibrate about this configuration, if dis- 
placed. The simplest case that can be proposed is when the 
surface is a cylinder of any form, and the equilibrium position 
of the string is perpendicular to the generating lines. The student 
will easily prove that the motion is independent of the curvature 
of the cylinder, and that the vibrations are in all essential respects 
the same as if the surface were developed into a plane. The case 
of an endless string, forming a necklace round the cylinder, is 
worthy of notice. 

In order to illustrate the characteristic features of this class of 
problems, we will take the comparatively simple example of a 
string stretched on the surface of a smooth sphere, and lying, 
when in equilibrium, along a great circle. The co-ordinates to 
which it will be most convenient to refer the s}rstem are the 
latitude 6 measured from the great circle as equator, and the 
longitude ^ measured along it. If the radius of the sphere be a, 
we have 

T=ljp(adyad<t>=^^je»d<t, (1). 

The extension of the string is denoted by 

Now 

ds" = (adey + (a cos d d<f>y ; 
so that 

Thus 
and 



-«/{(• 



d<^ (2); 



If the ends be fixed, 

^ Cambridge Mathematical Tripos Examination, 1876. 



214 TRANSVERSE VIBRATIONS OF STRINGS. [139. 

and the equation of virtual velocities is 

a'pf'esed<i>-'aT,j^Be(^+e)d<f>=o, 

whence, since Bd is arbitrary, 

«V^-2'«(|^. + ^) (3> 

This is the equation of motion. 
If we assume d qc cos pt, we get 

i + »-TV*-« W. 

of which the solution, subject to the condition that vanishes 
with 0, is 

0^Asm\-jjrp^ + l- 4>. cos pt (5). 

The remaining condition to be satisfied is that vanishes when 
o^ = Z, or (^ = a, if a = l/a. 

This gives 

T, /m«7r» \ r,/m'7r» 1\ 

^=aVV"^"V=7l"7^""a»j ^^>' 

where m is an integer. 

The normal functions are thus of the same form as for a 
straight string, viz. 

= A8Ui — cospt (/), 

but the series of periods is different. The effect of the curvature 
is to make each tone graver than the corresponding tone of a 
straight string. If a > tt, one at least of the values of />* is nega- 
tive, indicating that the corresponding modes are unstable. If 
a = IT, ^ is zero, the string being of the same length in the dis- 
placed position, as when d = 0. 

A similar method might be applied to calculate the motion of 
a string stretched round the equator of any surface of revolu- 
tion\ 

140. The approximate solution of the problem for a vibrating 
string of nearly but not quite uniform longitudinal density has been 
fully considered in Chapter iv. § 91, as a convenient example of 

^ [For a more general treatment of this question see MicheU, Messenger of 
Mathematics, vol. xzx. p. S7, 1890.] 



140.] VARIABLE DENSITY. 215 

the general theory of approximately simple systems. It will be 
sufficient here to repeat the result. If the density be po + Bp, the 
period t^ of the r^ component vibration is given by 

"■"f-{>na'='-7^i *» 

If the irregularity take the form of a small load of ma.<« m 
at the point x = b, the formula may be written 



-•=f'l'-S''-?l ■:■■''' 



These values of 7^ are correct as far as the first power of the 
small quantities Bp and m, and give the means of calculating a 
correction for such slight departures from uniformity as must 
always occur in practice. 

As might be expected, the effect of a small load vanishes at 
nodes, and rises to a maximum at the points midway between 
consecutive nodes. When it is desired merely to make a rough 
estimate of the effective density of a nearly uniform string, the 
formula indicates that attention is to be given to the neighbour- 
hood of loops rather than to that of nodes. 

[The effect of a small variation of density upon the period is 
the same whether it occur at a distance x from one end of the 
string, or at an equal distance from the other end. The mean 
variation at points equidistant from the centre is all that we need 
regard, and thus no generality will be lost if we suppose that the 
density remains symmetrically distributed with respect to the centre. 
Thus we may write 

T,» = ^"(l + ar) (3) 

where ar = T I — (1 — cos, jdx (4). 

In this equation Bp may be expanded from to ^Z in the series 

Bp A A 27ra? . . 27rrx . ,.. 

— = ^0 + ^1 cos—,-- + ... + -Ay cos — = h (o), 

Po I V 



where 

P 



Ao^^T-^d^ (6), 

Wo Po 



ilr = T I —COS f- dx (7). 

t.'o />0 ^ 



216 TRANSVERSE VIBRATIONS OF STRINGS. [l40. 

Accordingly, 

Or^A.^^Ar (8). 

This equation, as it stands, gives the changes in period in 
terms of the changes of density supposed to be known. And 
it shews conversely that a variation of density may always be 
found which will give prescribed arbitrary displacements to all 
the perioda This is a point of some interest. 

In order to secure a reasonable continuity in the density, it is 
necessaiy to suppose that a,, a, ••• are so prescribed that a,, assumes 
ultimately a constant value when r is increased indefinitely. If 
this condition be satisfied, we may take Ao = a^, and then Ar tends 
to zero as r increases. 

As a simple example, suppose that it be required so to vary 
the density of a string that, while the pitch of the fundamental 
tone is displaced, all other tones shall remain unaltered. The 
conditions give 

Accordingly 

and ill = — 2ai. 

Thus by (6) 

Bp/po = — 2ai cos {27ra!/l),] 

141. The differential equation determining the motion of a 
string, whose longitudinal density p is variable, is 



P 






dt^ 'da^ 



from which, if we assume y oc cosjp^, we obtain to determine the 
normal functions 

Z^^py-^ (2). 

where v^ is written for p^/Ti. This equation is of the second 
order and linear, but has not hitherto been solved in finite terms. 
Considered as defining the curve assumed by the string in the 
normal mode under consideration, it determines the curvature at 
any point, and accordingly embodies a rule by which the curve 
can be constructed graphically. Thus in the application to a 
string fixed at both ends, if we start fi:om either end at an arbitrary 



141.] VARIABLE DENSITY. 217 

« 

inclination, and with zero curvature, we are always directed by the 
equation with what curvature to proceed, and in this way we 
may trace out the entire curve. 

If the assumed value of v* be right, the curve will cross 
the axis of x at the required distance, and the law of vibration 
will be completely determined. If i^ be not known, different 
values may be tried until the curve ends rightly; a sufficient 
approximation to the value of v^ may usually be arrived at by a 
calculation founded on an assumed type (§§ 88, 90). 

Whether the longitudinal density be uniform or not, the 
periodic time of any simple vibration varies cceteria paribus as the 
square root of the density and inversely as the square root of the 
tension under which the motion takes place. 

The converse problem of determining the density, when the 
period and the type of vibration are given, is alwajrs soluble. For 
this purpose it is only necessary to substitute the given value of y, 
and of its second differential coefficient in equation (2). Unless 
the density be infinite, the extremities of a string are points of 
zero curvature. 

When a given string is shortened, every component tone is 
raised in pitch. For the new state of things may be regarded as 
derived from the old by introduction, at the proposed point of 
fixture, of a spring (without inertia), whose stiffness is gradually 
increased without limit. At each step of the process the potential 
energy of a given deformation is augmented, and therefore (§ 88) 
the pitch of every tone is raised. In like manner an addition to 
the length of a string depresses the pitch, even though the added 
part be destitute of inertia. 

142. Although a general integration of equation (2) of § 141 
is beyond our powers, we may apply to the problem some of the 
many interesting properties of the solution of the linear equation 
of the second order, which have been demonstrated by MM. Sturm 
and Liouville^ It is impossible in this work to give anything 
like a complete account of their investigations ; but a sketch, in 
which the leading features are included, may be found interesting, 
and will throw light on some points connected with the general 

^ The memoirs referred to in the text are contained in the first volume of 
Liooville's Journal (1S36). 



218 TRANSVEBSB VIBRATIONS OF STRINGS. [l42. 

theory of the vibrations of continuous bodies. I have not thought 
it necessary to adhere very closely to the methods adopted in the 
original memoira 

At no point of the curve satisf}dng the equation 

^^'^py-^ (1)' 

can both y and dyldx vanish together. By successive differen- 
tiations of (1) it is easy to prove that, if y and dy/dx vanish 
simultaneously, all the higher differential coefficients d^y/da:^, 
d^y/da^y &c. must also vanish at the same point, and therefore 
by Taylor's theorem the curve must coincide with the axis of x. 

Whatever value be ascribed to i/*, the curve satisfying (1) is 
sinuouSy being concave throughout towards the axis of x, since 
p is everywhere positive. If at the origin y vanish, and dy/dx 
be positive, the ordinate will remain positive for all values of x 
below a certain limit dependent on the value ascribed to v^. 
If J/* be very small, the curvature is slight, and the curve will 
remain on the positive side of the axis for a great distance. We 
have now to prove that as j/* increases, all the values of x which 
satisfy the equation y = gradually diminish in magnitude. 

Let y' be the ordinate of a second curve satisfying the equa- 
tion 

^'+'''Vy' = o (2). 

as well as the condition that y' vanishes at the origin, and let us 
suppose that v'^ is somewhat greater than i^. Multiplying (2) by 
y, and (1) by y', subtracting, and integrating with respect to x 
between the limits and x, we obtain, since y and y' both vanish 
with X, 

y't-y%-(-"-''^Jlpyy'^ <3)- 

If we further suppose that x corresponds to a point at which 
y vanishes, and that the difference between v'* and v* is very small, 
we get ultimately 

y'^=^^jjy'da! (4). 

The right-hand member of (4) being essentially positive, we 
learn that y' and dy/dx are of the same sign, and therefore that, 



142.] Sturm's thborem. 219 

whether dyjdx be positive or negative, j/ is already of the same 
sign as that to which y is changing, or in other words, the value 
of X for which y vanishes is less than that for which y vanishes. 

If we fix our attention on the portion of the curve lying 
between x = Q and a? = Z, the ordinate continues positive through- 
out as the value of i/* increases, until a certain value is attained, 
which we will call v^. The function y is now identical in form 
with the first normal function ttj of a string of density p fixed 
at and 2, and has no root except at those points. As i/* again 
increases, the first root moves inwards from x = l until, when a 
second special value v^ is attained, the curve again crosses the 
axis at the point x^l, and then represents the second normal 
function u^. This function has thus one internal root, and one 
only. In like manner corresponding to a higher value v^ we 
obtain the third normal function u^ with two internal roots, and 
so on. The v!-^ fiinction u^ has thus exactly n—\ internal roots, and 
since its first differential coefficient never vanishes simultaneously 
with the function, it changes sign each time a root is passed. 

From equation (3) it appears that if Ur and Ug be two different 
normal functions, 

pUrUgdx^O (5). 



J Q 



A beautiful theorem has been discovered by Sturm relating 
to the number of the roots of a function derived by addition 
from a finite number of normal functions. If Um be the component 
of lowest order, and tin the component of highest order, the function 

where ^, <f>m+i, &c. are arbitrary coefficients, has at least m— 1 
internal roots, and at most n — 1 internal roots. The extremities 
at a: = and At x = l correspond of course to roots in all cases. 
The following demonstration bears some resemblance to that given 
by Liouville, but is considerably simpler, and, I believe, not less 
rigorous. 

If we suppose that f(x) has exactly fi internal roots (any 
number of which may be equal), the derived function/" (a:) cannot 
have less than ^i + 1 internal roots, since there must be at least 
one root off (x) between each pair of consecutive roots of /(a?), and 
the whole number of roots of f(x) concerned is ^t -h 2. In like 
manner, we see that there must be at least fi roots of /"(a?), 



{ 



220 TRANSVEEISE VIBRATIONS OF STRINGS. [142. 

besides the extremities, which themselves necessarily correspond 
to roots; so that in passing from f{x) to f {x) it is impossible 
that any roots can be lost. Now 

= - P {Vrn (f>m W,n + V^m+i <f>m+i ^wi+i + + I'n' <^n ^). • (7), 

as we see by (1); and therefore, since p is always positive, we 
infer that 

^m' <^m ^^ + I^tiH-i <^m+i Wtn+l + ■^Vn4>n^n (8), 

has at least fi, roots. 

Again, since (8) is an expression of the same form as f(x), 
similar reasoning proves that 

has at least fi internal roots ; and the process may be continued 
to any extent. In this way we obtain a series of functions, all 
with fi internal roots at least, which differ from the original 
function /(re) by the continually increasing relative importance of 
the components of the higher orders. When the process has been 
carried sufficiently far, we shall arrive at a function, whose form 
differs as little as we please from that of the normal frinction of 
highest order, viz. it^^ and which has therefore n — 1 internal roots. 
It follows that, since no roots can be lost in passing down the 
series of functions, the number of internal roots of /(a?) cannot 
exceed n — 1. 

The other half of the theorem is proved in a similar manner 
by continuing the series of functions backwards from /(os). In 
this way we obtain 

VnT* ^m ^ + V^m+i <^ni+i l^+i + + ^'n"* ^ "„ 



•> 



arriving at last at a function sensibly coincident in form with the 
normal function of lowest order, viz. -m^, and having therefore 
m — 1 internal roots. Since no roots can be lost in passing up the 
series from this function to f(x), it follows that f(x) cannot have 
fewer internal roots than m — 1 ; but it must be understood that 
any number of the m — 1 roots may be equal. 

• We will now prove that f{x) cannot be identically zero, unless 



142.] EXPANSION IN SERIES OF NORMAL FUNCTIONS. 221 

all the coefficients ^ vanish. Suppose that <f}r is not zero. 
Multiply (6) by pUr, and integrate with respect to x between the 
limits and L Then by (5) 



I P'^f{^)dx = if>r\ pur^dx (9); 

JO Jo 



from which, since the integral on the right-hand side is finite, we 
see that f(x) cannot vanish for all values of x included within the 
range of integration. 

Liouville has made use of Sturm's theorem to shew how a 
series of normal functions may be compounded so as to have an 
arbitrary sign at all points lying between x^O and x=^l. His 
method is somewhat as follows. 

The values of x for which the function is to change sign being 
a, 6, c, ..., quantities which without loss of generality we may 
suppose to be all different, let us consider the series of determi- 
nants, 

1*1 (a), Ui{x) I I ?ti(a), Wi(6), tii{x) 

Ma (a), Vni^) \ I Vii((l)y ^3(6), lUiix) 

th (a), "9 (6), fh (^) i , &c. 

The first is a linear function o( v^(x) and Ui(x), and by Sturm's 
theorem has therefore one internal root at most, which root is 
evidently a. Moreover the determinant is not identically zero, 
since the coefficient of w, (x), viz. i^ (a), does not vanish, whatever 
be the value of a. We have thus obtained a function, which 
changes sign at an arbitrary point a, and there only internally. 

The second determinant vanishes when x = a, and when ^r = 6, 
and, since it cannot have more than two internal roots, it changes 
sign, when x passes through these values, and there only. The 
coefficient of u^ (x) is the value assumed by the first determinant 
when re = 6, and is therefore finite. Hence the second determinant 
is not identically zero. 

Similarly the third determinant in the series vanishes and 
changes sign when x = a, when x = b, and when x = c, and at these 
internal points only. The coefficient of U4(x) is finite, being the 
value of the second determinant when x = c. 

It is evident that by continuing this process we can form 
functions compounded of the normal functions, which shall vanish 
and change sign for any arbitrary values of x, and not elsewhere 



222 TRANSVERSE VIBRATIONS OF STRINGS. [142. 

internally ; or, in other words, we can form a function whose sign 
is arbitrary over the whole range from x = to x^l. 

On this theorem Liouville founds his demonstration of the 
possibility of representing an arbitrary function between x = and 
J? = Z by a series of normal functions. If we assume the possibility 
of the expansion and take 

/(a?) = ^ii^(a?) + 0,2i,(a?) + 0,i^,(a:) + (10), 

the necessary values of ^i, <^a, &c. are determined by (9), and we 
find 

f(x) = ^\nr(x)l pUr(x)/(x)dx-7-\ pv^^(x)dx> (11). 

If the series on the right be denoted by F(x), it remains to 
establish the identity o{f(x) and F(x), 

If the right-hand member of (11) be multiplied by pur(x) and 
integrated with respect to x from x = to x — l,we see that 

I ptlr{x)F{x)dx—j ptCr(x)f(x)dx, 

Jo Jo 

or, as we may also write it, 

{F(x)''f(x)}pur(x)dx = (12), 



Jc 





where Ur(x) is any normal function. From (12) it follows that 



/, 



{F(x) -f(x)} [Aiu, (x) + il,w, (x) + A^v^(x)+. . .} p (ic = 0. . .(13), 



where the coefficients Ai, A^, &c. are arbitrary. 

Now if F(x) -f(x) be not identically zero, it will be possible 
so to choose the constants A^, A^, &c. that AiUi (a;) + ilst^(^)+ ... 
has throughout the same sign as F{x)—/(x\ in which case every 
element of the integral would be positive, and equation (13) could 
not be true. It follows that F(x)—f(x) cannot differ from zero, 
or that the series of normal frmctions forming the right-hand 
member of (11) is identical with /(a?) for all values of x from x = 
to a? = i. 

The arguments and results of this section are of course ap- 
plicable to the particular case of a uniform string for which the 
normal functions are circular. 

[As a particular case of variable density the supposition that 



142.] VARIABLE DENSITY. 223 

p = aar^ is worthy of notice, § 148 6. In the notation there 
adopted 

m«-hi = 7i«=p»cr/2\ (14), 

and the general solution is 

y = Aa^'^+Ba:^"' (15). 

If the string be fixed at two points, whose abscissae Xi, x^ are 
as r to 1, the frequency equation is r^"* =» 1, or 

"'^i+dgvy w 

where 8 denotes an integer. The proper frequencies thus depend 
only upon the ratio of the terminal abscissae. By supposing r 
nearly equal to unity we may fall back upon the usual formula 
(§ 124) applicable to a uniform string. 

The general form of the normal function is 



1 . sir log (xlx^ ., ^. T 



142 a. The points where the string remains at rest, or nodes, 
are of course determined by the roots of the normal functions, 
when the vibrations are free. In this case the frequency is limited 
to certain definite values ; but when the vibrations are forced, they 
may be of any frequency, and it becomes possible to trace the 
motion of the nodal points as the frequency increases continuously. 

For example, suppose that the imposed force acts at a single 
point P of a string AB, whose density may be variable. So long 
as the frequency is less than that of either of the two parts AP, 
PB (supposed to be held at rest at both extremities) into which 
the string is divided, there can be no (interior) node (Q). Other- 
wise, that part of the string AQ between the node Q and one 
extremity (-4), which does not include P, would be vibrating 
freely, and more slowly than is possible for the longer length AP, 
included between the point P and the same extremity. When the 
frequency is raised, so as to coincide with the smaller of those 
proper to AP, PB, say ilP, a node enters at P and then advances 
towards A. At each coincidence of the frequency with one of 
those proper to the whole string AB, the vibration identifies itself 
with the corresponding free vibration, and at each coincidence with 
a frequency proper to AP, or BP, a new node appears at P, and 



224 TRANSVERSE VIBRATIONS OF STRINGS. [142 a. 

advances in the first case towards A and in the second towards B. 
And throughout the whole sequence of events all the nodes move 
outwards from P towards A or B. 

Thus, if the string be uniform and be bisected at P, there is 
no node until the pitch rises to the octave (c') of the note (c) of the 
string. At this stage two nodes enter at P, and move outwards 
sjrmmetrically. When g' is reached, the mode of vibration is that 
of the fi^ee vibration of the same pitch, and the nodes are at the 
two points of trisection. At c" these nodes have moved outwards 
so far as to bisect AP, BP, and two new nodes enter at P. 

143. When the vibrations of a string are not confined to one 
plane, it is usually most convenient to resolve them into two sets 
executed in perpendicular planes, which may be treated inde- 
pendently. There is, however, one case of this description worth 
a passing notice, in which the motion is most easily conceived and 
treated without resolution. 



Suppose that 



. 8irx 2sirt 
V = sm — s- cos 

. sirx . 2s7rt 
z = sm —V- sm 

i T 



r 



(1). 



Then 



STTX 



r = V(y^-h^«) = 8in ^- (2), 

and z : y = tan (2«7r^/T) (3), 

shewing that the whole string is at any moment in one plane, 
which revolves uniformly, and that each particle describes a circle 
with radius sin (swx/l). In fact, the whole system turns without 
relative displacement about its position of equilibrium, completing 
each revolution in the time r/s. The mechanics of this case is 
quite as simple as when the motion is confined to one plane, the 
resultant of the tensions acting at the extremities of any small 
portion of the string's length being balanced by the centrifugal 
force. 

144. The general differential equation for a uniform string, 
viz. 

^y = a'^ (1) 



144.] UNLIMITKD STRING. 225 

may be transformed by a change of variables into 

& = <2). 

where u^x — at,v = X'\'at. The general solution of (2) is 

y^f{u) + F(v)=f(a^-'at)-\'F(x + at) (3)\ 

f, F being two arbitrary functions. 

Let us consider first the case in which F vanishes. When 
t has any particular value, the equation 

y^f{x-^at) (4). 

expressing the relation between x and y, represents the form of the 
string. A change in the value of ^ is merely equivalent to an 
alteration in the origin of x, so that (4) indicates that a certain 
f(yrm is propagated along the string with uniform velocity a in the 
positive direction. Whatever the value of y may be at the point 
X and at the time ty the same value of y will obtain at the point 
x + aMdki the time t -h A^. 

The form thus perpetuated may be any whatever, so long as it 
does not violate the restrictions on which (1) depends. 

When the motion consists of the propagation of a wave in the 
positive direction, a certain relation subsists between the inclina- 
tion and the velocity at any point. Differentiating (4) we find 

S=-«£ » 

Initially, dyldt and dyjdx may both be given arbitrarily, but if 
the above relation be not satisfied, the motion cannot be repre- 
sented by (4). 

In a similar manner the equation 

y^Fix-^at) (6) 

denotes the propagation of a wave in the negative direction, and 
the relation between dyjdt and dyjdx corresponding to (5) is 

i-«i <'> 

In the general case the motion consists of the simultaneous 
propagation of two waves with velocity a, the one in the positive, 

1 [Bqofttions (1) and (8) are due to D'Alembert (1750).] 
R. 15 



I 



226 



TRANSVERSE VIBRATIONS OP STRINGS. 



[144. 



and the other in the negative direction; and these waves are 
entirely independent of one another. In the first dyjdt = — a dy/dx, 
and in the second dyldt — adyjdx. The initial values of rfy/ctt 
and dyjdx must be conceived to be divided into two parts, which 
satisfy respectively the relations (5) and (7). The first constitutes 
the wave which will advance in the positive direction without 
change of form ; the second, the negative wave. Thus, initially, 



/»-r(.)=-l| 



whence 



(8). 



equations which determine the functions /' and F' for all values 
of the argument from a? = — oo to a? = oo , if the initial values of 
dyjdx and dyjdt be known. 

If the disturbance be originally confined to a finite portion of 
the string, the positive and negative waves separate after the 
interval of time required for each to traverse half the disturbed 
portion. 

Fig. 28. 



Suppose, for example, that AB is the part initially disturbed. 
A point P on the positive side remains at rest until the positive 
wave has travelled from -4 to P, is disturbed during the passage 
of the wave, and ever after remains at rest. The negative wave 
never affects P at all. Similar statements apply, mviaUs mutandis, 
to a point Q on the negative side of AB, If the character of the 
original disturbance be such that a dyjdx — dyjdt vanishes initially, 
there is no positive wave, and the point P is never disturbed at 
all ; and if a dyjdx + dyjdt vanish initially, there is no negative 
wave. If dyjdt vanish initially, the positive and the negative 
waves are similar and equal, and then neither can vanish. In 
cases where either wave vanishes, its evanescence may be con- 
sidered to be due to the mutual destruction of two component 



144.] POSITIVE AND NEGATIVE WAVES. 227 

waves, one depending on the initial displacements, and the other 
on the initial velocities. On the one side these two waves con- 
spire, and on the other they destroy one another. This explains 
the apparent paradox, that P can fail to be affected sooner or later 
after AB has been disturbed. 

The subsequent motion of a string that is initially displaced 
without velocity, may be readily traced by graphical methods. 
Since the positive and the negative waves are equal, it is only 
necessary to divide the original disturbance into two equal parts, 
to displace these, one to the right, and the other to the left, 
through a space equal to at, and then to recompound them. We 
shall presently apply this method to the case of a plucked string 
of finite length. 

146. Vibrations are called stationary, when the motion of each 
particle of the system is proportional to some function of the time, 
the same for all the particles. If we endeavour to satisfy 

t"'^ <•)• 

by assuming y = XT, where X denotes a function of x only, and 
T a function of t only, we find 

Td^'^Xda?''^ (a constant), 
SO that 



T = -4 cos mat -h B sin mat 
X = (7 cos WW? + D sin mx 



] (2). 



proving that the vibrations must be simple harmonic, though of 
arbitrary period. The value of y may be written 

y = P COS {mat — e) cos {ma — a) 
-\Pqo& {mxii + ma: — 6 — a) + iPcos {ma;t — !?«; — € + a).. .(3), 

shewing that the most general kind of stationary vibration may 
be regarded as due to the superposition of equal progressive vibra- 
tions, whose directions of propagation are opposed. Conversely, 
two stationary vibrations may combine into a progressive one. 

The solution y=f{x — at) + F{x-¥at) applies in the first 
instance to an infinite string, but may be interpreted so as to 
give the solution of the problem for a finite string in certain 

15—2 



228 TRANSVERSE VIBRATIONS OF STRINGS. [l45. 

casea Let us suppose, for example, that the string terminates 
at x = 0, and is held fast there, while it extends to infinity in 
the positive direction only. Now so long as the point ^r = 
actually remains at rest, it is a matter of indifference whether 
the string be prolonged on the negative side or not. We are 
thus led to regard the given string as forming part of one doubly 
infinite, and to seek whether and how the initial displacements 
and velocities on the negative side can be taken, so that on 
the whole there shall be no displacement at a; = throughout the 
subsequent motion. The initial values of y and y on the positive 
side determine the corresponding parts of the positive and negative 
waves, into which we know that the whole motion can be resolved. 
The former has no influence at the point x = 0. On the negative 
side the positive and the negative waves are initially at our dis- 
posal, but with the latter we are not concerned. The problem is 
to determine the positive wave on the negative side, so that in 
conjunction with the given negative wave on the positive side 
of the origin, it shall leave that point undisturbed. 

Let OPQRS... be the line (of any form) representing the 
wave in OX, which advances in the negative direction. It is 




evident that the requirements of the case are met by taking on 
the other side of what may be called the contrary wave, so that 
is the geometrical centre, bisecting every chord (such as PP') 
which passes through it. Analytically, if y =/(a?) is the equation 

of OPQRS , -y^fi-x) is the equation of OFQ'RS: 

When after a time t the curves are shifted to the left and to 
the right respectively through a distance at, the co-ordinates 
corresponding to a; = are necessarily equal and opposite, and 
therefore when compounded give zero resultant displacement. 

The effect of the constraint at may therefore be represented 



145.] 



REFLECTION AT A FIXED POINT. 



229 



by supposing that the negative wave moves through undisturbed, 
but that a positive wave at the same time emerges from 0. This 
reflected wave may at any time be found from its parent by the 
following rule : 

Let APQRS.,. be the position of the parent wave. Then the 
reflected wave is the position which this would assume, if it were 

Fig. 25. 





turned through two right angles, flrst about OX as an axis of 
rotation, and then through the same angle about OY. In other 
words, the return wave is the image of APQRS formed by 
successive optical reflection in OX and OT, regarded as plane 
mirrors. 

The same result may also be obtained by a more analytical 
process. In the general solution 

y =/(aj - at) -h F(x + at), 

the functions /(-gr), F{z) are determined by the initial circumstances 
for all positive values of z. The condition at a; = requires that 

for all positive values of t, or 

/(- z) = - Fiz) 

for positive values of z. The functions / and F are thus de- 
termined for all positive values of x and t 

There is now no difficulty in tracing the course of events when 
ttvo points of the string A and B are held fast. The initial dis- 
turbance in AB divides itself into positive and negative waves, 
which are reflected backwards and forwards between the fixed 
points, changing their character from positive to negative, and 
vice versd, at each reflection. After an even number of reflec- 
tions in each case the original form and motion is completely 



230 



TRANSVERSE VIBRATIONS OF STRINGS. 



[145. 



recovered. The process is most easily followed in imagination 
when the initial disturbance is confined to a small pcui) of the 
string, more particularly when its character is such as to give rise 
to a wave propagated in one direction only. The pulse travels with 
uniform velocity (a) to and firo along the length of the string, and 
after it has returned a second time to its starting point the 
original condition of things is exactly restored. The period of 
the motion is thus the time required for the pulse to traverse 
the length of the string twice, or 

T = 2Z/a (1). 

The same law evidently holds good whatever may be the character 
of the original disturbance, only in the general case it may 
happen that the shortest period of recurrence is some aliquot part 

of T. 

146. The method of the last few sections may be advantage- 
ously applied to the case of a plucked string. Sinc6 the initial 
velocity vanishes, half of the displacement belongs to the positive 
and half to the negative wave. The manner in which the wave 
must be completed so as to produce the same effect as the con- 
straint, is shewn in the figure, where the upper curve represents 

]^. 26. 





the positive, and the lower the negative wave in their initial 
positions. In order to find the configuration of the string at any 
future time, the two curves must be superposed, after the upper 
has been shifted to the right and the lower to the left through a 
space equal to o^. ^ 



146. J GRAPHICAL METHOD. 231 

The resultant curve, like its components, is made up of straight 
piecea A succession of six at intervals of a twelfth of the period, 

Fig. 27. 








shewing the course of the vibration, is given in the figure (Fig. 27), 
taken from Helmholtz. From the string goes back again to A . 
through the same stages \ 

It will be observed that the inclination of the string at the 
points of support alternates between- two constant values. 

« 

147. If a small disturbance be made at the time t at the 
point X of an infinite stretched string, the effect will not be felt 
at until after the lapse of the time a/a, and will be in all 
respects the same as if a like disturbance had been made at 
the point sb + Ax at time t — Ax/a: Suppose that similar dis- 
turbances are communicated to the string at intervals of time 
r at points whose distances from increase each time by'aSr, 
then it is evident that the result at.O will be the same as if the 
disturbances were all made at the same point, provided that the 
time-intervals be increased from t to t + St. This remark con- 

^ This method of treating the vibration of a plooked string is dae to Tooag. 
Phil. Trant.t 1800. The student is recommended to make himself familiar with it 
by aotoaUy oonstmoting the forms of Fig. 27. 



232 TRANSVEBSE VIBRATIONS OF STRINGS. [l47. 

tains the theory of the alteration of pitch due to motion of the 
source of disturbance ; a subject which will come under our notice 
again in connection with aerial vibrations. 

148. When one point of an infinite string is subject to a forced 
vibration, trains of waves proceed from it in both directions ac- 
cording to laws, which are readily investigated. We shall suppose 
that the origin is the point of excitation, the string being there 
subject to the forced motion y = Ae^^\ and it will be sufficient to 
consider the positive side. If the motion of each element ds be 
resisted by the frictional force Kpyds, the differential equation is 

S-I--S c)^ 

or since y oc e*'^, 

^'(!f-^^-y (»)■ 

if for brevity we write \^ for the coefficient of y. 

The general solution is 

y = {(7e-^» + D e+^} e»i^ (3). 

Now since y is supposed to vanish at an infinite distance, D 
must vanish, if the real part of X be taken positive. Let 

where a is positive. 

Then the solution is 

y = ^e-<*+*>*+»^ (4), 

or, on throwing away the imaginary part, 

y-Ae-** cos(pt-l3x) (5), 

corresponding to the forced motion at the origin 

y = A cos pt (6). 

An arbitrary constant may, of course, be added to t 

To determine a and /3, we have 

)^ 
a*' a* 

If we suppose that k is small, 

/3 = p/a, a = K/2a nearly. 



«.»^ = -g; 2a/8==^ (7). 



and 



y = Ae-'^f^co&(pt-^x\ (8). 



148.] DAMPING OF PROGRESSIVE WAVES. 233 

This solution shews that there is propagated along the string 
a wave, whose amplitude slowly diminishes on account of the 
exponential factor. If k = 0, this factor disappears, and wo have 
simply 

y = ^ cos [pt-^) (9). 

This result stands in contradiction to the general law that, 
when there is no friction, the forced vibrations of a system (due 
to a single simple harmonic force) must be s}aichronous in phase 
throughout. Accoixling to (9), on the contrary, the phase varies 
continuously in passing from one point to another along the string. 
The fact is, that we are not at liberty to suppose ie = in (8), 
inasmuch as that equation was obtained on the assumption that 
the real part of \ in (3) is positive, and not zero. However long 
a finite string may be, the coefficient of friction may be taken so 
small that the vibrations are not damped before reaching the 
further end. After this point of smallness, reflected waves begin 
to complicate the result, and when the friction is diminished 
indefinitely, an infinite series of such must be taken into account, 
and would give a resultant motion of the same phase throughout. 

This problem may be solved for a string whose mass is supposed 
to be concentrated at equidistant points, by the method of § 120. 
The co-ordinate '^i may be supposed to be given {=n&^*\ and 
it will be found that the system of equations (5) of § 120 may all 
be satisfied by taking 

tr = ^^ti (10), 

where is a complex constant determined by a quadratic equa- 
tion. The result for a continuous string may be afterwards 
deduced. 

[In the notation of § 120 the quadratic equation is 

5^ + ^^+5 = (11), 

where il=-^/)=--f -\ B=--^ (12). 

The roots of (11) are 

^- 25 ^^^'' 

and are imaginary if 45" > A*, that is, if 



P'<^ <!*)• 



234 TRANSVERSE VIBRATIONS OF STRINGS. [148. 

a condition always satisfied in passing to the limit where a and /^ 
are infinitely small. In any case when (14) is satisfied the 
modulus of ^ is unity, so that (10) represents wave propagation. 

If, however, (14) be not satisfied, the values of are real. In 
this case all the motions are in the same phase, and no wave 
is propagated. The vibration impressed upon ^i is imitated upon 

a reduced scale by -^a* V^s > with amplitudes which form a 

geometrical progression. In the first case the motion is pro- 
pagated to an infinite distance, but in the second it is practically 
confined to a limited region round the source.] 

148 a. So long as the conditions of § 144 are satisfied, a 
positive, or a negative, wave is propagated undisturbed. If 
however there be any want of uniformity, such (for example) as 
that caused by a load attached at a particular point, reflection 
will ensue when that point is reached. The most interesting 
problem under this head is that of two strings of diflFerent 
longitudinal densities, attached to one another, and vibrating 
transversely under the common tension 2\. Or, if we regard the 
string as single, the density may be supposed to vary dia- 
continuously from one uniform value (p,) to another (p,). If 
Oi, Oj denote the corresponding velocities of propagation, 

a,«= TiK a,« = 2V/>, (1), 

and /A = Oi/oa = V W/>i) (2). 

The conditions to be satisfied at the junction of the two parts 
are (i) the continuity of the displacement y, and (ii) the continuity 
of dyldx. If the two parts met at a finite angle, an infinitely 
small element at the junction would be subject to a finite force. 

Let us suppose that a positive wave of harmonic type, travelling 
in the first part (p,), impinges upon the second (/>,). In the latter 
the motion will be adequately represented by a positive wave, 
but in the former we must provide for a negative reflected wave. 
Thus we may take for the two parts respectively 

y = fie*»<»»<-^+ire*» <«»*+«> (3), 

y=Ze*«<««*-^ (4), 

where ki = 27r/Xi , k^ = 27r/X, , 

so that kidi^k^ (5). 



148 a.] REFLECTION AT A JUNCTION. 285 

The conditions at the junction (a?= 0) give 

H-¥K^L (6), 

k,H^k,K = kJ. (7); 

, K ki-ki /A-1 .Qv 

whence n^^i^ — r = — ^- — ? (p)- 

Since the ratio K/H is real, we may suppose that both 
quantities are real; and if we throw away the imaginary parts 
from (3) and (4) we get as the solution in terms of real quantities 

y ss H cos ki{(iit '- x) •{- K coaki (oit •\' x) (9); 

y^(H + K) cos k^(aj;-x) (10). 

The ratio of amplitudes of the reflected and the incident 
waves expressed by (8) is that first obtained by T. Young for 
the corresponding problem in Optics. 

148 b. The expression for the intensity of reflection established 
in § 148 a depends upon the assumption that the transition from 
the one density to the other is sudden, that is occupies a distance 
which is small in comparison with a wave length. If the 
transition be gradual, the reflection may be expected to fall off, 
and in the limit to disappear altogether. 

The problem of gradual transition includes, of course, that of 
a variable medium, and would in general be encumbered with 
great difficulties. There is, however, one case for which the 
solution may be readily expressed, and this it is proposed to 
consider in the present section. The longitudinal density is 
supposed to vary as the inverse square of the abscissa. If y, 
denoting the transverse displacement be proportional to ^^, the 
equation which it must satisfy as a function o{ x, is (§ 141), 

g+n«^-*y = (1). 

where n* is some positive constant, of the nature of an abstract 
number. 

The solution of (1) is y = Ax^+'''' + Bx^*^ (2), 

where 71%^ = ^,^ — ^ (3). 

If m be real, that is, if n > i, we may obtain, by supposing 
il = 0, as a final solution in real quantities, 

y = CWcos (pt — mlog X + e) (4), 



• 



236 TRANSVERSE VIBRATIONS OF STRINGS. [148 6. 

which represents a positive progressive wave, in many respects 
similar to those propagated in uniform media. 

Let us now suppose that, to the left of the point x^Xi, the 
variable medium is replaced by one of uniform constitution, such 
that there is no discontinuity of density at the point of transition ; 
and let us inquire what reflection a positive progressive wave in 
the uniform medium will undergo on arrival at the variable 
medium. It will be sufficient to consider the case where m is 
real, that is, where the change of density is but moderately rapid. 

By supposition, there is no negative wave in the variable 
medium, so that -4 = in (2). Thus 

and, when ic = a?i, ? =■- (5). 

ydx Xi ^ 

The general solution for the uniform medium, satisfying the 
equation dhfjda? + n^Xi~^y = 0, may be written 

y = He '• +Ke *» (6), 

from which, when x = x,, 

dy^ _ _in H - K .,. 

ydx" x,H + K ^ ^' 

In equation (6), H represents the amplitude of the incident 
positive wave, and K the amplitude of the reflected negative 
wave. The condition to be satisfied at x=^x^ is expressed by 

equating the values of —J; given by (5) and (7). Thus 

if 

which gives, in symbolical form, the ratio of the reflected to the 
incident vibration. 

Having regard to (3), we may write (8) in the form 

ir"'2(n+m) ^^' 

so that the amplitude of the reflected wave is J(n + m)""^ of 
that of the incident. Thus, as was to be expected, when n and m 
are great, t.e., when the density changes slowly in the variable 



I48 6.] GRADUAL TRANSITION. 237 

medium, there is but little reflection. As regards phase, the 
result embodied in (9) may be represented by supposing that the 
reflection occurs at a; = a:i, and involves a change of phase amount- 
ing to a quarter period. 

Passing on now to the more important problem, we will 
suppose that the variable medium extends only so far as the point 
x^Xi, beyond which the density retains uniformly its value at 
that point. A positive wave travelling at first in a uniform 
medium of density proportional to xr^, passes at the point a? = a?i 
into a variable medium of density proportional to ar^, and again, at 
the point x = x^, into a uniform medium of density proportional to 
Xi~^, The velocities of propagation are inversely proportional to 
the square roots of the densities, so that, if ft be the refractive 
index between the extreme media, 

Xt 

M = ^ (10). 

The thickness (d) of the layer of transition is 

d = a?s-ar, (11). 

The wave-lengths in the two media are given by 

_ 2irxi __ iirxj 

\i — — - , A, — ~ -— - ; 

fi n 

2ird 27rd 
so that « = x,--x, = 0*---i)X. (12). 

For the first medium we take, as before, 

y = He '» -^-Ke '^ (6), 

giving, when x^Xi, 

dy __ inH — K _ _^ inO .,^ 

ydx^ XiH+K^ Xi ^ ^' 

if, for brevity, we write for „ -fr • 

For the variable medium, 

y = ilici+'"* + fiari-*'" (2), 

giving, when x = Xi, 

dy__^_,(^ + im) Ax^^ + (i - im) Bxr'" .. «. 

ydx"' Ax.^^-BxT'^ ^^'^^' 



238 TRANSYEBSB VIBRATIONS OF STBINQS. [148 6. 

Hence the condition to be satisfied tA x = Xi gives 

whence ^ = ^.-«.»^^^| (14). 

The condition to be satisfied at x^x^ may be deduced from (14), 
by substituting x^ for x^, putting at the same time ^ = 1 in virtue 
of the supposition that in the second medium there is no negative 
wave. Hence, equating the two values oi A:B, we get 

Xi-^"^ -. — —T-w—-^ ^xf^'^- — —, f- (15), 

as the equation from which the reflected wave in the first medium 
is to be found. Having regard to (3), we get 

~ H-hK - rn + n -li^pL^^(m -^ n + Ji) ' 

80 that ^= Q-. \ -.-cT^^/ \ (16). 

H 2 (m + n) + 2/a**» (m — n) ^ "^ 

This is the symbolical solution. To interpret it in real quantities, 
we must distinguish the cases of m real and m imaginary. If the 
transition be not too sudden, m is real, and (16) may be written 

K _i — 1 + cos (2m log /a) + i sin (2m log /a) 

JT "" 2 m + n + (m — w) cos (2m log /a) + i (m — n) sin (2m log /a) ' 

Thus the expression for the ratio of the intensities of the reflected 
and the incident waves is, after reduction, 

sinHmlog /i) 

4m' + sin* (m log /a) ^ ^' 

If m be imaginary, we may write im^m' \ (16) then gives for 
the ratio of intensities, 

or, if we introduce the notation of hyperbolic trigonometry § 170, 

sinh^m^log/^) ,,qv 

sinh« (m' log /i) + 4m'» ^ ^' 

For the critical value m = 0, we get, from (17) or (19), 

4 + (log^)« ^^^'- 



148 6.J GRADUAL TRANSITION. 289 

These expressions allow us to trace the effect of a more or 
less gradual transition between media of given indices. If the 
transition be absolutely abrupt, n = 0, by (12); so that m' = i. 
In this case, (18) gives us (§ 148 a) Young's well-known formula 

> - 1\' 



U+i-' 



.(21). 



smh (Cm 
Since increases continually from x^O, the ratio (19) 

X 

increases continually from m=0 to m'«=J, i.e., diminishes 
continually frt)m the case of sudden transition m' = ^, when its 
value is (21), to the critical case m = 0, when its value is (20), 
after which this form no longer holds good. When m' =^0,n^\, 
and, by (12), d = (X, - \)/ Anr. 

When n>\, (17) is the appropriate form. We see from it 
that with increasing n the reflection diminishes, until it vanishes, 
when mlog/i = 7r, i.e. when 

"-*+(i^ •• • <*^>- 

With a still more gradual transition the reflection revives, reaches 
a maximum, again vanishes when m log /i » 27r, and so on^ 

148 c. In the problem of connected strings, vibrating imder 
the influence of tension alone, the velocity in each uniform part is 
independent of wave length, and there is nothing corresponding to 
optical dispersion. This state of things will be departed frx>m if 
we introduce the consideration of stiffness, and it may be of interest 
to examine in a simple case how far the problem of reflection is 
thereby modified. As in § 148 a, we will suppose that at a; = 
the density changes discontinuously from />i to p,, but that now 
the vibrations of the second part occur under the influence of 
sensible stiffness. The differential equation applicable in this 
case is, § 188, 



or, if y vary as e**^, 






-'^S + «'"S+"'^ = (!>• 

SO that, if y vsjj as e***, 

^A-* + a^»i'»-n>=0 (2). 

1 Proc. Math, Soc,, vol. xi. Febmary, 18S0 ; where will also be found a numeri- 
cal example illustrative of optical conditions. 



240 TRANSVERSE VIBRATIONS OF STRINGS. [148 C. 

In consequence of the stiflftiess represented by ^ the velocity 
of propagation deviates from a,, and must be found from (2). The 
two values of k^ given by this equation are real, one being positive 
and the other negative. The four admissible values of k may thus 
be written + k^y ± ih^, so that the complete solution of (1) will be 

y = Ae^^ + Be-^^-¥Ce-'^ + D^ (3), 

/ts» k^ being real and positive. The velocity of propagation is n/k^ 

In the application which we have to make the disturbance of 
the imperfectly flexible second part is due to a positive wave 
entering it from the first part. When x is great and positive, (3) 
must reduce to its second term. Thus 

^=0, Z) = 0; 

and we are left with 

y = £^-<M + (7^M (4). 

This holds when x is positive. When x is negative, corresponding 
to the perfectly flexible first part, we have 

y = JETe-**'* + JfiTc*** (5), 

in which ki=^n/ai (6). 

The " refractive index " is given by 

fi^kjk, (7). 

The conditions at the junction are first the continuity of y and 
dyjdx. Further, d^y/da^ in (4) must vanish at this place, inasmuch 
as curvature implies a couple (§ 162), and this could not be 
transmitted by the first part. Hence 

H+K=B+C (8), 

h{H-K)^k^''ihjO (9), 

-&,»5 + A,'C=:0 (10). 

From these we deduce 

H-K~ kji, ^ ^' 

K^_hj{ki — kt) + iktKi .^ _. _ 

H~h{k, + k,) + ikA ^ ^' 



148 c.] IMPERFECT FLEXIBILITY. 241 

and thence for the intensity of reflection, equal to Mod^. (K/H\ 

\iCi ~" "'i) I '•'1 '•'2 /"'3 



(k, + k,y + k\%'/hj^ 



(13). 



If the second part, as well as the first, be perfectly flexible, 
^ = 0, Aj '= 00 , and we fall back on Young's formula. In general, 
the intensity of reflection is not accurately given by this formula, 
even though we employ therein the value of the refractive index 
appropriate to the waves actually under propagation. 



R. 16 



CHAPTER VII. 

LONGITUDINAL AND TORSIONAL VIBRATIONS OF BARS. 

149. The next system to the string in order of simplicity 
is the bar, by which term is usually understood in Acoustics a 
mass of matter of uniform substance and elongated cylindrical 
form. At the ends the cylinder is cut off by planes perpendicular 
to the generating lines. The centres of inertia of the transverse 
sections lie on a straight line which is called the cuds. 

The vibrations of a bar are of three kinds — longitudinal, 
torsional, and lateral. Of these the last are the most important, 
but at the same time the most diflScult in theory. They are 
considered by themselves in the next chapter, and will only be 
referred to here so far as is necessary for comparison and contrast 
with the other two kinds of vibrations. 

Longitudinal vibrations are those in which the axis remains 
unmoved, while the transverse sections vibrate to and fro in the 
direction perpendicular to their planes. The moving power is 
the resistance offered by the rod to extension or compression. 

One peculiarity of this class of vibrations is at once evident. 
Since the force necessary to produce a given extension in a bar 
is proportional to the area of the section, while the mass to be 
moved is also in the same proportion, it follows that for a bar of 
given length and material the periodic times and the modes of 
vibration are independent of the area and of the form of the 
transverse section. A similar law obtains, as we shall presently 
see, in the case of torsional vibrations. 

It is otherwise when the vibrations are lateral. The periodic 
times are indeed independent of the thickness of the bar in the 
direction perpendicular to the plane of flexure, but the motive power 



149.] CLASSIFICATION OF VIBRATIONS. 243 

in this case, viz. the resistance to bending, increases more rapidly 
than the thickness in that plane, and therefore an increase in 
thickness is accompanied by a rise of pitch. 

In the case of longitudinal and lateral vibrations, the mechan- 
ical constants concerned are the density of the material and the 
value of Young's modulus. For small extensions (or compressions) 
Hooke's law, according to which the tension varies as the extension, 

, ,, , T- ., . . . actual lenirth — natural lencfth 

holds eood. If the extension, viz. ^ ^, rr — , 

® natural length 

be called e, we have T=q€, where q is Young's modulus, and T 
is the tension per unit area necessary to produce the extension €. 
Young's modulus may therefore be defined as the force which would 
have to be applied to a bar of unit section, in order to double its 
length, if Hooke's law continued to hold good for so great exten- 
sions ; its dimensions are accordingly those of a force divided by an 
area. 

The torsional vibrations depend also on a second elastic con- 
stant fly whose interpretation will be considered in the proper 
place. 

Although in theory the three classes of vibrations, depending 
respectively on resistance to extension*, to torsion, and to flexure 
are quite distinct, and independent of one another so long as the 
squares of the strains may be neglected, yet in actual experiments 
with bars which are neither uniform in material nor accurately 
cylindrical in figure it is often found impossible to excite longi- 
tudinal or torsional vibrations without the accompaniment of some 
measure of lateral motion. In bars of ordinary dimensions the 
gravest lateral motion is far graver than the gravest longitudinal 
or torsional motion, and consequently it will generally happen that 
the principal tone of either of the latter kinds agrees more or less 
perfectly in pitch with some overtone of the former kind. Under 
such circumstances the regular modes of vibrations become 
unstable, and a small irregularity may produce a great effect. The 
difficulty of exciting purely longitudinal vibrations in a bar is 
similar to that of getting a string to vibrate in one plane. 

With this explanation we may proceed to consider the three 
classes of vibrations independently, commencing with longitudinal 
vibrations, which will in fact raise no mathematical questions 
beyond those already disposed of in the previous chapters. 

16—2 



244 LONGITUDINAL VIBRATIONS OF BARS. [150. 

150. When a rod is stretched by a force parallel to its length, 
the stretching is in general accompanied by lateral contraction in 
such a manner that the augmentation of volume is less than if 
the displacement of every particle were parallel to the axis. In the 
case of a short rod and of a particle situated near the cylindrical 
boundary, this lateral motion would be comparable in magnitude 
with the longitudinal motion, and could not be overlooked without 
risk of considerable error. But where a rod, whose length is great 
in proportion to the linear dimensions of its section, is subject 
to a stretching of one sign throughout, the longitudinal motion 
accumulates, and thus in the case of ordinary rods vibrating 
longitudinally in the graver modes, the inertia of the lateral 
motion may be neglected. Moreover we shall see later how a 
correction may be introduced, if necessary. 

Let X be the distance of the layer of particles composing any 
section from the equilibrium position of one end, when the rod 
is unstretched, either by permanent tension or as the result of 
vibrations, and let f be the displacement, so that the actual 
position is given by x + (. The equilibrium and actual position 

of a neighbouring layer being x-i- Sx, a; + &» + f + -7^ S^ re- 
spectively, the elongation m d^jdx, and thus, if T be the tension 
per unit area acting across the section, 

^=^f (!)• 

Consider now the forces acting on the slice bounded by x 
and X'\-hx, If the area of the section be co, the tension at a; is 
by (1) qtod^/dx, acting in the negative direction, and at x + Bx 
the tension is 

acting in the positive direction; and thus the force on the slice 
due to the action of the adjoining parts is on the whole 

The mass of the element is pay &r, if p be the original density, 
and therefore if X be the accelerating force acting on it, the 
equation of equilibrium is 

^+?2-» «■ 



150.] GENERAL DIFFERENTIAL EQUATION. 245 

In what follows we shall not require to consider the operation 
of an impressed force. To find the equation of motion we have 
only to replace X by the reaction against acceleration — ^, and 
thus if 5 : p = a*, we have 

dC " dx' ^^^- 

This equation is of the same form as that applicable to the 
transverse displacements of a stretched string, and indicates the 
undisturbed propagation of waves of any tjrpe in the positive and 
negative directions. The velocity a is relative to the unstretched 
condition of the bar ; the apparent velocity with which a disturb- 
ance is propagated in space will be greater in the ratio of the 
stretched and unstretched lengths of any portion of the bar. The 
distinction is material only in the case of permanent tension. 

161. For the actual magnitude of the velocity of propagation, 

we have 

a^ = q : p = qoi) : pco, 

which is the ratio of the whole tension necessary (according to 
Hooke's law) to double the length of the bar and the longitudinal 
density. If the same bar were stretched with total tension T, 
and were flexible, the velocity of propagation of waves along it 
would be \/(y • P«)- I*^ order then that the velocity might be 
the same in the two cases, T must be qo), or, in other words, the 
tension would have to be that theoretically necessary in order to 
double the length. The tones of longitudinally vibrating rods 
are thus very high in comparison with those obtainable from 
strings of comparable length. 

In the case of steel the value of q is about 22 x 10* grammes 
weight per square centimetre. To express this in absolute units 
of force on the c. G.s.* system, we must multiply by 980. In 
the same system the density of steel (identical with its specific 
gravity referred to water) is 7*8. Hence for steel 

/980 X 2 2 X 10^ r,«^oAA 
a = y ^^ = 530,000 

approximately, which shews that the velocity of sound in steel is 
about 530,000 centimetres per second, or about 16 times greater 

^ Centimetre, Gramme, Second. This system is recommended by a Committee 
of the British Association. Brit, Aas. Report, 1878. 



246 LONGITUDINAL VIBRATIONS OF BARS. [151. 

than the velocity of sound in air. In glass the velocity is about 
the same as in steel. 

It ought to be mentioned that in strictness the value of q deter- 
mined by statical experiments is not that which ought to be used 
here. As in the case of gases, which will be treated in a subsequent 
chapter, the rapid alterations of state concerned in the propaga- 
tion of sound are attended with thermal effects, one result of 
which is to increase the effective value of q beyond that obtained 
from observations on extension conducted at a constant tempera- 
ture. But the data are not precise enough to make this correction 
of any consequence in the case of solids. 

162. The solution of the general equation for the longitudinal 
vibrations of an unlimited bar, namely 

^=f{x-at)-¥F{x^-at\ 

being the same as that applicable to a string, need not be further 
considered here. 

When both ends of a bar are free, there is of course no perma- 
nent tension, and at the ends themselves there is no temporary 
tension. The condition for a free end is therefore 

§-» m. 

To determine the normal modes of vibration, we must assume 
that f varies as a harmonic function of the time^-cos nat Then 
as a function of x, f must satisfy 

S+~'^=' <2>- 

of which the complete integral is 

f = -4 cos rue + fisinwa? (.3), 

where A and B are independent of x. 

Now since d^jdx vanishes always when x = 0, we get 5 = 0; and 
again since d^/dx vanishes when x = I — the natural length of the 
bar, sin nl = 0, which shews that n is of the form 






% being integral. 



152.] BOTH EXTREMITIES FREE. 247 

Accordingly, the normal modes are given by equations of the 
form 

-, , iirx iirat ,^. 

f = -4 cos -,- cos— y— (o), 

in which of course an arbitrary constant may be added to t, if 
desired. 

The complete solution for a bar with both ends free is there- 
fore expressed by 



(6), 



f=2^^cos ^ |il<cos-j- + 5iSin— p •... 

where Ai and Bi are arbitrary constants, which may be determined 
in the usual manner, when the initial values of f and ^ are 
given. 

A zero value of % is admissible ; it gives a term representing a 
displacement f constant with respect both to space and time, 
and amounting in fact only to an alteration of the origin. 

The period of the gravest component in (6) corresponding to 
i = 1, is 2f/a, which is the time occupied by a disturbance in 
travelling twice the length of the rod. The other tones found 
by ascribing integral values to i form a complete harmonic scale ; 
so that according to this theory the note given by a rod in 
longitudinal vibration would be in all cases musical. 

In the gravest mode the centre of the rod, where x = Ji, is a 
place of no motion, or node ; but the periodic elongation or com- 
pression d^/dx is there a maximum. 

163. The case of a bar with one end free and the other fixed 
may be deduced from the general solution for a bar with both 
ends free, and of twice the length. For whatever may be the 
initial state of the bar free at a? = and fixed at a: = Z, such dis- 
placements and velocities may always be ascribed to the sections 
of a bar extending from to 21 and free at both ends as shall 
make the motions of the parts from to i identical in the two 
cases. It is only necessary to suppose that from I to 21 the dis- 
placements and velocities are initially equal and opposite to those 
found in the portion from to Z at an equal distance from the 
centre x =^L Under these circumstances the centre must by 
the symmetry remain at rest throughout the motion, and then the 



248 LONGITUDINAL VIBRATIONS OF BARS. [153. 

portion from to Z satisfies all the required conditions. We con- 
clude that the vibrations of a bar free at one end and fixed at the 
other are identical with those of one half of a bar of twice the 
length of which both ends are free, the latter vibrating only in the 
uneven modes, obtained by making i in succession all odd integers. 
The tones of the bar still belong to a harmonic scale, but the 
even tones (octave, &c. of the fundamental) are wanting. 

The period of the gravest tone is the time occupied by a pulse 
in travelling /cmr times the length of the bar. 

164. When both ends of a bar are fixed, the conditions to 
be satisfied at the ends are that the value of ^ is to be invariable. 
At a? = 0, we may suppose that f = 0. At a: = f, f is a small 
constant a, which is zero if there be no permanent tension. In- 
dependently of the vibrations we have evidently f = a? a -r- Z, and 
we should obtain our result most simply by assuming this term 
at once. But it may be instructive to proceed by the general 
method. 

Assuming that as a function of the time f varies ss 

A cos nat + B sin nat, 
we see that as a function of x it must satisfy 

of which the general solution is 

f = (7 cos nx + D^iu nx (1). 

But since f vanishes with x for all values of ^, G = 0, and thus 
we may write 

f = 2 sin Tia? }-4 cos nat + B sin ncU], 

The condition at a; = Z now gives 

2 sin nl [A cos nat -|- 5 sin nat] = a, 

from which it follows that for every finite admissible value of n 

7 A *^ 

sm nZ = 0, or n = -r- . 

But for the zero value of w, we get 

Aq sin nl = a, 



154.] BOTH EXTREMITIES FIXED. 249 

and the corresponding term in f is 

t = Ao sin nx = a — , = a r • 

* sin nl I 

The complete value of f is accordingly 



f = «7 + A.j sm j- 



, iirat -n ' ifrat 

Ai cos -, — h Bi sm —j- 



...(2). 



The series of tones form a complete harmonic scale (from 
which however any of the members may be missing in any 
actual case of vibration), and the period of the gravest com- 
ponent is the time taken by a pulse to travel twice the length 
of the rod, the same therefore as if both ends were free. It 
must be observed that we have here to do with the unstretched 
length of the rod, and that the period for a given natural length 
is independent of the permanent tension. 

The solution of the problem of the doubly fixed bar in the 
case of no permanent tension might also be derived from that 
of a doubly free bar by mere differentiation with respect to a. 
For in the latter problem d^/dx satisfies the necessary differential 
equation, viz. 

dt^ \dx) da? \dx 

inasmuch as ^ satisfies 

and at both ends d^jdx vanishes. Accordingly d^fdv in this 
problem satisfies all the conditions prescribed for f in the case 
when both ends are fixed. The two series of tones are thus 
identical. 

155. The effect of a small load if attached to any point of 
the rod is readily calculated approximately, as it is sufficient 
to assume the type of vibration to be unaltered (§ 88). We 
will take the case of a rod fixed at a? = 0, and free At x = l. The 
kinetic energy is proportional to 



Jo 



p(t) sm* ^y clx + Jif Sin' . 



pcol /_ 2M . „ iirxX 



250 LONGITUDINAL VIBRATIONS OF BARS. [l55. 

Since the potential energy is unaltered, we see by the prin- 
ciples of Chapter iv., that the effect of the small load AT at a 
distance x fr(5m the fixed end is to increase the period of the 
component tones in the ratio 

The small quantity M : ptol is the ratio of the load to the 
whole mass of the rod. 

If the load be attached at the free end, sin'(i7rir/2Z) = l, and 
the effect is to depress the pitch of every tone by the same small 
interval. It will be remembered that i is here an uneven integer. 

If the point of attachment of M be a node of any component, 
the pitch of that component remains unaltered by the addition. 

156. Another problem worth notice occurs when the load at 
the free end is great in comparison with the mass of the rod. 
In this case we may assume as the type of vibration, a condition 
of uniform extension along the length of the rod. 

If f be the displacement of the load M, the kinetic energy is 

.0 ^ 

The tension corresponding to the displacement f is qco f/Z, 
and thus the potential energy of the displacement is 

Fv?*"-? (2) 

The equation of motion is 

(M + ip<ol) ? + ^^^ f = 0, 
and if f X cos pt 

P' = ^^{M + ifxol) (3). 

The correction due to the inertia of the rod is thus equivalent 
to the addition to 3f of one-third of the mass of the rod. 

156 tt. So long as a rod or a wire is uniform, waves of longi- 
tudinal vibration are propagated along it without change of type, 
but any interruption, or alteration of mechanical properties, will 
in general give rise to reflection. If two uniform wires be joined, 



156 a. J CORRECTION FOR LATERAL MOTION. 251 

the problem of determining the reflection at the junction may be 
conducted as in § 148 a. The conditions to be satisfied at the 
junction are (i) the continuity of f, and (ii) the continuity of 
qcjd^fdx, measuring the tension. If pi, /o,, ©i, ©a, Oi, a^ denote 
the volume densities, the sections, and the velocities in the two 
wires, the ratio of the reflected to the incident amplitude is 
given by 

H pitOiOi ■\- p^io^o^ 

The reflection vanishes, or the incident wave is propagated 
through the junction without loss, if 

pi<0\(ti = p^cc^a^ (2). 

This result illustrates the diflSculty which is met with in obtaining 
effective transmission of sound from air to metal, or from metal to 
air, in the mechanical telephone. Thus the value of pa is about 
100,000 times greater in the case of steel than in the case of air. 

157. Our mathematical discussion of longitudinal vibrations 
may close with an estimate of the error involved in neglecting 
the inertia of the lateral motion of the parts of the rod not 
situated on the axis. If the ratio of lateral contraction to longi- 
tudinal extension be denoted by fi, the lateral displacement of a 
particle distant r from the axis will be fire in the case of equili- 
brium, where € is the extension. Although in strictness this 
relation will be modified by the inertia of the lateral motion, yet 
for the present purpose it may be supposed to hold good, § 88. 

The constant /a is a numerical quantity, Ij'ing between and ^. 
If fjL were negative, a longitudinal tension would produce a lateral 
swelling, and if fi were greater than ^, the lateral contraction 
would be great enough to overbalance the elongation, and cause 
a diminution of volume on the whole. The latter state of things 
would be inconsistent with stability, and the former can scarcely 
be possible in ordinary solids. At one time it was supposed 
that fjL was necessarily equal to J, so that there was only one 
independent elastic constant, but experiments have since shewn 
that /JL is variable. For glass and brass Wertheim found experi- 
mentally fi = ^. 

If f) denote the lateral displacement of the particle distant r 



252 LONGITUDINAL VIBRATIONS OP BARS. [157. 

from the axis, and if the section be circular, the kinetic energy 
due to the lateral motion is 

Thus the whole kinetic energy is 

In the case of a bar free at both ends, we have 

iirx df iir . iirx 

^'^'^^ I ' ^*-T«^° i • 

and thus 

The effect of the inertia of the lateral motion is therefore to 
increase the period in the ratio 

This correction will be nearly insensible for the graver modes of 
bars of ordinary proportions of length to thickness. 

[A more complete solution of the problem of the present 
section has been given by Pochhammer^ who applies the general 
equations for an elastic solid to the case of an infinitely extended 
cylinder of circular section. The result for longitudinal vibrations, 
so far as the term in r^jl^, is in agreement with that above deter- 
mined. A similar investigation has also been published by Chree', 
who has also treated the more general question' in which the 
cylindrical section is not restricted to be circular.] 

158. Experiments on longitudinal vibrations may be made 
with rods of deal or of glass. The vibrations are excited by 
friction § 138, with a wet cloth in the case of glass; but for metal 
or wooden rods it is necessary to use leather charged with powdered 
rosin. " The longitudinal vibrations of a pianoforte string may be 
excited by gently rubbing it longitudinally with a piece of india 
rubber, and those of a violin string by placing the bow obliquely 
across the string, and moving it along the string longitudinally, 
keeping the same point of the bow upon the string. The note is 
unpleasantly shrill in both cases." 

1 CrelU, Bd. 81, 1876. ^ Quart, Math, Joum., Vol. 21, p. 287, 188C. 

» Ihid, Vol. 23, p. 317, 1889. 



158.] TORSIONAL VIBRATIONS. 253 

" If the peg of the violin be turned so as to alter the pitch of 
the lateral vibrations very considerably, it will be found that the 
pitch of the longitudinal vibrations has altered very slightly. The 
reason of this is that in the case of the lateral vibrations the 
change of velocity of wave-transmission depends chiefly on the 
change of tension, which is considerable. But in the cajse of the 
longitudinal vibrations, the change of velocity of wave-transmis- 
sion depends upon the change of extension, which is comparatively 
slight \" 

In Savart's experiments on longitudinal vibrations, a peculiar 
sound, called by him a " son rauque," was occasionally observed, 
whose pitch was an octave below that of the longitudinal vibra- 
tion. According to Terquem * the cause of this sound is a trans- 
verse vibration, whose appearance is due to an approximate 
agreement between its own period and that of the sub-octave of 
the longitudinal vibration § 68 b. If this view be correct, the 
phenomenon would be one of the second order, probably referable 
to the fact that longitudinal compression of a bar tends to produce 
curvature. 

159. The second class of vibrations, called torsional, which 
depend on the resistance opposed to twisting, is of very small 
importance. A solid or hollow cylindrical rod of circular section 
may be twisted by suitable forces, applied at the ends, in such a 
manner that each transverse section remains in its own plane. 
But if the section be not circulai-, the effect of a twist is of a 
more complicated character, the twist being necessarily attended 
by a warping of the layers of matter originally composing the 
normal sections. Although the effects of the warping might pro- 
bably be determined in any particular case if it were worth 
while, we shall confine ourselves here to the case of a circular 
section, when there is no motion parallel to the axis of the rod. 

The force with which twisting is resisted depends upon an 
elastic constant different from q, called the rigidity. If we de- 
note it by n, the relation between g, n, and fi may be written 

"=20.Vl) <^>' 

^ Donkin*8 Acoustic*, p. 154. 
« Ann. de Chimie, lvii. 129—190. 

' Thomsou and Tait, § 683. This, it should be remarked, applies to isotropic 
material only. 



254 LONGITUDINAL VIBRATIONS OF BARS. [159. 

shewing that n lies between J 5 and J5. In the case of /a= J, 

Let us now suppose that we have to do with a rod in the form 
of a thin tube of radius r and thickness dr, and let d denote the 
angular displacement of any section, distant x from the origin. 
The rate of twist at a; is represented by dd/dx, and the shear of the 
material composing the pipe by r dO/dx, The opposing force per 
unit of area is nr dO/dx ; and since the area is 2irr dr, the moment 
round the axis is 

27171-7^ dr -J- . 
dx 

Thus the force of restitution acting on the slice dx has the 
moment 

271717^ dr dx i— , 

Now the moment of inertia of the slice under consideration 
is 2'7rrdr.dx.p.r^j and therefore the equation of motion assumes 
the form 

d^0 d^e 

P^^^^'d^ (2>- 

Since this is independent of r, the same equation applies to a 
cylinder of finite thickness or to one solid throughout. 

The velocity of wave propagation is '\/(n/p), and the whole 
theory is precisely similar to that of longitudinal vibrations, the 
condition for a free end being dd/dx = 0, and for a fixed end ^ = 0, 
or, if a permanent twist be contemplated, d = constant. 

The velocity of longitudinal vibrations is to that of torsional 
vibrations in the ratio V? • V'^ or V(2 + 2/a) : 1. The same ratio 
applies to the frequencies of vibration for bars of equal length 
vibrating in corresponding modes under corresponding terminal 
conditions. If /a = J, the ratio of frequencies would be 

V?:\/w = \/8 : V3 = l*63, 

corresponding to an interval rather greater than a fifth. 

In any case the ratio of frequencies must lie between 

V2 : 1 = 1-414, and V3 : 1 = 1*732. 

Longitudinal and torsional vibrations were first investigated by 
ChladnL 



CHAPTER VIII. 

LATERAL VIBRATIONS OF BARS. 

160. In the present chapter we shall consider the lateral 
vibrations of thin elastic rods, which in their natural condition are 
straight. Next to those of strings, this class of vibrations is per- 
haps the most amenable to theoretical and experimental treatment. 
There is difficulty sufficient to bring into prominence some im- 
portant points connected with the general theory, which the fami- 
liarity of the reader with circular functions may lead him to pass 
over too lightly in the application to strings ; while at the same 
time the difficulties of analysis are not such as to engross attention 
which should be devoted to general mathematical and physical 
principles. 

Daniel Bernoulli ' seems to have been the first who attacked 
the problem. Euler, Riccati, Poisson, Cauchy, and more recently 
Strehlke', Lissajous", and A. Seebeck* are foremost among those 
who have advanced our knowledge of it. 

161. The problem divides itself into two parts, according to 
the presence, or absence, of a permanent longitudinal tension. 
The consideration of permanent tension entails additional compli- 
cation, and is of interest only in its application to stretched 
strings, whose stiffness, though small, cannot be neglected al- 
together. Our attention will therefore be given principally to the 
two extreme cases, (1) when there is no permanent tension, 
(2) when the tension is the chief agent in the vibration. 

1 Comment, Acad, Petrop, t. xiii. * Pogg. Ann, Bd. xxvii. p. 505, 1833. 

* Ann. d, Chimie (3), xxx. 385, 1850. 

* Ahhandlungen d. Math. Phys. Clatte d. K. Sdchs. QetelUehaft d. Wissen- 
9cha/ten. Leipzig, 1852. 



256 LATERAL VIBRATIONS OF BARS. [l61. 

With respect to the section of the rod, we shall suppose that 
one principal axis lies in the plane of vibration, so that the bending 
at every part takes place in a direction of maximum or minimum 
(or stationary) flexural rigidity. For example, the surface of the 
rod may be one of revolution, each section being circular, though 
not necessarily of constant radius. Under these circumstances the 
potential energy of the bending for each element of length is pro- 
portional to the square of the curvature multiplied by a quantity 
depending on the material of the rod, and on the moment of 
inertia of the transverse section about an axis through its centre of 
inertia perpendicular to the plane of bending. If o) be the area 
of the section, /c'© its moment of inertia, q Young's modulus, ds the 
element of length, and dV the corresponding potential energy for 
a curvature 1 -r- ii of the axis of the rod, 

dr=ig«=a,g (1). 

This result is readily obtained by considering the extension of 
the various filaments of which the bar may be supposed to be 
made up. Let 77 be the distance from the axis of the projection 
on the plane of bending of a filament of section d(o. Then the 
length of the filament is altered by the bending in the ratio 

R being the radius of curvature. Thus on the side of the axis for 
which 17 is positive, viz. on the otUward side, a filament is extended, 
while on the other side of the axis there is compression. The 
force necessary to produce the extension rj/R is q rj/R . dto by the 
definition of Young's modulus; and thus the whole couple by 
which the bending is resisted amounts to 



jq^,rj,d(o = ^K^<o, 



if CD be the area of the section and k its radius of g3rration about 
a line through the axis, and perpendicular to the plane of bending. 
The angle of bending corresponding to a length of axis ds 18 ds-i-JR, 
and thus the work required to bend ds to curvature 1 -r i2 is 

ds 
^' 

since the mean is half the final value of the couple. 

[For a more complete discussion of the legitimacy of .the 



Jg/c*© 



161.] POTENTIAL ENERGY OP BENDING. 257 

foregoing method of calculation the reader must be referred to 
works upon the Theory of Elasticity. The question of lateral 
vibrations has been specially treated by Pochhammer^ on the 
basis of the general equations.] 

For a circular section ic is one-half the radius. 

That the potential energy of the bending would be proportional, 
C(Bte7^ paribtM, to the square of the curvature, is evident before- 
hand. If we call the coefficient B^ we may take 



-iK- 



or, in view of the approximate straightness, 



=i/-©^ (^). 



in which y is the lateral displacement of that point on the axis of 
the rod whose abscissa, measured parallel to the undisturbed posi- 
tion, is X. In the case of a rod whose sections are similar and 
similarly situated i^ is a constant, and may be removed from under 
the integral sign. 

The kinetic energy of the moving rod is derived partly from 
the motion of translation, parallel to y, of the elements composing 
it, and partly from the rotation of the same elements about axes 
through their centres of inertia perpendicular to the plane of vibra- 
tion. The former part is expressed by 



^jfXD^cUc (3), 



if p denote the volume-density. To express the latter part, we have 
only to observe that the angular displacement of the element dx is 
dyjdx, and therefore its angular velocity (Py/dt dx. The square of 
this quantity must be multiplied by half the moment of inertia of 
the element, that is, by ^/^p€o dx. We thus obtain 

T=ifpcoil'dx + ^fK'pa>{^^Jdx (4). 

1 CrelU, Bd. 81, 1876. 
R. 17 



258 LATERAL VIBRATIONS OP BABS. [162. 

162. In order to form the equation of motion we may avail 
ourselves of the principle of virtual velocities. If for simplicity we 
confine ourselves to the case of uniform section, we have 

where the terms free from the integral sign are to be taken between 
the limits. This expression includes only the internal forces due 
to the bending. In what follows we shall suppose that there are 
no forces acting from without, or rather none that do work upon 
the system. A force of constraint, such as that necessary to hold 
any point of the bar at rest, need not be regarded, as it does no 
work and therefore cannot appear in the equation of virtual velo- 
cities. 

The virtual moment of the accelerations is 
Thus the variational equation of motion is 

in which the terms free from the integral sign are to be taken 
between the limita From this we derive as the equation to be 
satisfied at all points of the length of the bar 

while at each end 

or, if we introduce the value of B, viz. qi^m, and write qjp = b", 



162.] TERMINAL CONDITIONS. 259 

and for each end 

'•SKl)^{li-'"S}«^=« ('^ 

In these equations b expresses the velocity of transmission of 
longitudinal waves. 

The condition (5) to be satisfied at the ends assumes different 
forms according to the circumstances of the case. It is possible to 
conceive a constraint of such a nature that the ratio S {dyjdx) : hy 
has a prescribed finite value. The second boundary condition is 
then obtained from (5) by introduction of this ratio. But in all 
the cases that we shall have to consider, there is either no constraint 
or the constraint is such that either h {dyjdx) or hy vanishes, and 
then the boundary conditions take the form 

SK^-«' {^-"t]^-' <«>• 

We must now distinguish the special cases that may arise. If 
an end be free, iy and B(dy/dx) are both arbitrary, and the 
conditions become 

a^"""' d^dx ^ da?"^ ^^^' 

the first of which may be regarded as expressing that no couple 
acts at the free end, and the second that no force acta 

If the direction at the end be free, but the end itself be con- 
strained to remain at rest by the action of an applied force of the 
necessary magnitude, in which case for want of a better word the 
rod is said to be supported, the conditions are 



3-^ = 0, 



da^ 



Sy = (3), 



by which (5) is satisfied. 

A third case arises when an extremity is constrained to main- 
tain its direction by an applied couple of the necessary magnitude, 
but is free to take any position. We have then 

17—2 



260 LATERAL VIBRATIONS OF BARS. [162. 

Fourthly, the extremity may be constrained both as to 
position and direction, in which case the rod is said to be clamped. 
The conditions are plainly 



(|)=o. 8y=o (10). 



Of these four cases the first and the last are the more 
important; the third we shall omit to consider, as there are 
no experimental means by which the contemplated constraint 
could be realized. Even with this simplification a considerable 
variety of problems remain for discussion, as either end of the 
bar may be free, clamped or supported, but the complication 
thence arising is not so great as might have been expected. 
We shall find that difiFerent cases may be treated together, 
and that the solution for one case may sometimes be derived 
immediately from that of another. 

In experimenting on the vibrations of bars, the condition 
for a clamped end may be realized with the aid of a vice of 
massive construction. In the case of a free end there is of course 
no difiSculty so far as the end itself is concerned ; but, when both 
ends are free, a question arises as to how the weight of the bar 
is to be supported. In order to interfere with the vibration 
as little as possible, the supports must be confined to the neigh- 
bourhood of the nodal points. It is sometimes sufiicient merely 
to lay the bar on bridges, or to pass a loop of string round the hsjr 
and draw it tight by screws attached to its ends. For more exact 
purposes it would perhaps be preferable to carry the weight of 
the bar on a pin traversing a hole drilled through the middle of 
the thickness in the plane of vibration. 

When an end is to be 'supported,' it may be pressed inta 
contact with a fixed plate whose plane is perpendicular to the 
length of the bar. 

163. Before proceeding further we shall introduce a sup- 
position» which will greatly simplify the analysis, without seriously 
interfering with the value of the solution. We shall assume that 
the terms depending on the angular motion of the sections of 
the bar may be neglected, which amounts to supposing the 
inertia of each section concentrated at its centre. We shall 
afterwards (§ 186) investigate a correction for the rotatory in- 



163.] HARMONIC VIBRATIONS. 261 

ertia, and shall prove that under ordinary circumstances it is 
small. The equation of motion now becomes 

j'--»-s=» «• 

and the boundary conditions for a fi*ee end 

d^~^' da?-^ ^^^• 

The next step in conformity with the general plan will be 
the assumption of the harmonic form of y. We may conveniently 
take 

y=if cos/ -ii'niH\ (3), 

where I is the length of the bar, and m is an abstract number, 
whose value has to be determined. Substituting in (1), we 
obtain 

di* = -F" ^*>- 

If w = eP»»*/' be a solution, we see that p is one of the fourth 
roots of unity, viz. +1, —1, +i, — i; so that the complete 
solution is 

u=^ A coam-j+B sinm^ -{- C e"^'^ '\' D e"^^'^ (4a), 

containing four arbitrary constants. 

[The simplest case occurs when the motion is strictly periodic 
with respect to x, C and D vanishing. If X be the wave-length 
and T the period of the vibration, we have 

27r _ m 27r _ , m' 

so that T=^ , (46).] 

27r/co 

In the case of a finite rod we have still to satisfy the four 
boundary conditions, — two for each end. These determine the 
ratios A : B : C : D, and furnish besides an equation which m 
must satisfy. Thus a series of particular values of m are alone 
admissible, and for each m the corresponding u is determined in 
ever}rthing except a constant multiplier. We shall distinguish the 
different functions u belonging to the same system by suffixes. 



262 LATERAL VIBRATIONS OF BARS. [l63. 

The value of y at any time may be expanded in a series of 
the functions u (§§ 92, 93). If ^i, ^, &c. be the normal co- 
ordinates, we have 

y=<^i^i + «^M,+ (5), 

and T^^pw |(^iMi + ^2i^a + ...)*da; 

= ipft)|<^f [t/i»c;tr+<^//i/,»AF+... • (6). 

We are fully justified in asserting at this stage that each 
integrated product of the functions vanishes, and therefore the 
process of the following section need not be regarded as more 
than a verification. It is however required in order to determine 
the value of the integrated squares. 

164. Let w,», Um! denote two of the normal functions cor- 
responding respectively to m and m\ Then 

d*u^ _m\^ d*u„,' _ m!^ 

1^^ l*^' (ir* " l*"^"' ^^^' 

or, if dashes indicate differentiation with respect to (mx/l), 

t^"" = t^n, Um''''^Um' (2). 

If we subtract equations (1) after multiplying them by u,„', 
«^ respectively, and then integrate over the length of the bar, 
we have 

_ d^Um' d^Un, dUm' d^Um du^d^U„,' , 

"^"*"d^^ ''''•' da^^ dx dx' dx dx' ^'^^' 

the integrated terms being taken between the limits. " 

Now whether the end in question be clamped, supported, or 
free^ each term vanishes on account of one or other of its 

^ The reader should observe that the cases here specified are particular, and 
that the right-hand member of (3) vanishes, provided that 

"• • da;' "* ' dar» ' 

and ^^ : ^!!^* = ^' : ^^' , 

dx ' dx* dx ' dx^ 

These conditions include, for instance, the case of a rod whose end is urged 
towards its position of equilibrium by a force proportional to the displacement, as 
by a spring without inertia. 



164.] CONJUGATE PROPERTY. 263 

fisK^tors. We may therefore conclude that, if Um> ^' refer to two 
modes of vibration (corresponding of course to the same terminal 
conditions) of which a rod is capable, then 



/• 



hl,nUni'dx = (4). 

provided m and wi' be different. 

The attentive reader will perceive that in the process just 
followed, we have in fact retraced the steps by which the funda- 
mental differential equation was itself proved in § 162. It is the 
original variational equation that has the most immediate con- 
nection with the conjugate property. If we denote yhyu and &y 

and the equation in question is 

^jd^^^+p''h'^=^ <^)- 

Suppose now that u relates to a normal component vibration, 
so that il + n'w = 0, where n is some constant ; then 

d^u cPv 






By similar reasoning, if v be a normal function, and u represent 
any displacement possible to the system, 






We conclude that if u and v be both normal functions, which 
have different periods, 

vvda^O (6); 



/' 



and this proof is evidently as direct and general as could be 
desired. 

The reader may investigate the formula corresponding to (6), 
when the term representing the rotatory inertia is retained. 

By means of (6) we may verify that the admissible values of n* 
are real. For if w- were complex, and u^a+ifi were a normal 
function, then a - i^, the conjugate of w, would be a normal 
function also, corresponding to the conjugate of n", and then the 



264 LATERAL VIBRATIONS OF BARS. [l64. 

product of the two functions, being a sum of squares, would not 
vanish, when integrated^ 

If in (3) m and m be the same, the equation becomes iden- 
tically true, and we cannot at once infer the value of ju^^dx. 
We must take m' equal to m + Sm, and trace the limiting form of 
the equation as 5m tends to vanish. [It should be observed that 
the function Um-^^n is not a normal function of the system ; it is 
supposed to be derived from i/^ by variation of m in (4a) § 163, 
the coefficients A, B,C, D being retained constant.] In this way 
we find 

Z* J dmda^ dmda^ da^dmdx dxdmdx^* 

the right-hand side being taken between the limits. 

^T du m , o du X , o 

Now ^ = jti, &c., ^-=jw,&c., 



and thus 

^' f • J 3m' f„ m' 



4m' f . , 3m' ,„ rri?x ,,,, m^x , ,,, 
— *" ' uu — 74 ^^ 



/-./" 



, m* , ,, , ??i'a? , „.. 2m' , „ rn^x , 



in which w"" = w, so that 



-J \Un?dx = 



3uu'" + '^ V? - ^^ uu- ^ v:v;' -h ^ {y:y, . .(7), 



between the limits. 

Now whether an end be clamped, supported, or free, 

and thus, if we take the origin of x at one end of the rod. 



j ?*'da; = ]|(w«-2tA'?r-hu"')- 



= iZ(w'-2u'f/"-hw"%.i (8). 

The form of our integral is independent of the terminal con- 
dition at 0? = 0. If the end a? = i be free, u" and u'" vanish, and 
accordingly 

\ u^dx = Uu^l) (9), 



.'0 

^ This method is, I believe, doe to Poisson. 



164.] VALUES OF INTEGRATED SQUARES. 265 

that is to say, for a rod with one end free the mean value of t^' is 
one-fourth of the terminal value, and that whether the other end 
be clamped, supported, or free. 

Again, if we suppose that the rod is clamped Bit x = l, u and it' 
vanish, and (8) gives 



/: 



Since this must hold good whatever be the terminal condition at 
the other end, we see that for a rod, one end of which is fixed and 
the other free, 



/, 



u^dx ^\lu^ (free end) = \lu''^ (fixed end), 





shewing that in this case u^ at the free end is the same as u''^ at 
the clamped end. 

The annexed table gives the values of four times the mean of u^ 
in the different cases. 



I 

clamped, free i v? (free end), or w"* (clamped end) 

free, free | w' (free end) 

I 

clamped, clamped ... I u"^ (clamped end) 

supported, supported j — 2u'u'" (supported end) = 2m'' 

supported, free ; v? (free end), or - 2u'u'" (supported end) 

supported, clamped j u'*^ (clamped end), or - 2u'u'" (supported end) 



By the introduction of these values the expression for T 
assumes a simpler form. In the case, for example, of a clamped- 
free or a free-free rod, 

3f'=^{<^'t^'(0 + <^''^'(0 + ...} (10), 

where the end x=^l\% supposed to be free. 

165. A similar method may be applied to investigate the 
values of fu'^dx, and ju"^dx. In the derivation of equation (7) of 
the preceding section nothing was assumed beyond the truth of 
the equation u"** = u, and since this equation is equally true of any 



266 LATERAL VIBRATIONS OF BARS. [l65, 



of the derived functions, we are at liberty to replace u by u or u". 
Thus 

taken between the limits, since the term uv!' vanishes in all three 
cases. 

For a free-free rod 

^ J u'^dx = 3 (uu')i - 3 (mw')o + ^'i (^'0« 

= 6(ww')« + m(it'0« (1), 

for, as we shall see, the values of u u' must be equal aud opposite 
at the two ends. Whether u be positive or negative at a? = Z, 
u xi! is positive. 



For a rod which is clamped at a? = and free eX x — l 
4m 





Lm C^ 



[We have already seen that ?/o" = ± ui; and it may be proved 
from the formulse of § 173 that 

1//" _ tio' _ cos 7n + cosh m 
ui vi sin m sinh m ' 

xi_ X (w"^"')o (cos m + cosh mV , ^ 

so that \ . . ■ = - — ^"i . , a = - 1] 

(u u)i sm* m smh' m 

Thus ^^ j u'^dx==2(uu')i + mu{^ (2), 

a result that we shall have occasion to use later. 

By applying the same equation to the evaluation of / u'^dx, we 



find 



4m f 

Ti" 



"'dx = 3m"«' + ^ «"» -2'^ u'" u' - u'" u + y u' 



=rni(tt"»- 2m' «'" + «'),, 
since u'u" and uu'" vanish. 



165.] NORMAL EQUATIONS. 267 

Comparing this with (8) § 164, we see that 



h^-h''^ (3). 



whatever the terminal conditions may be. 

The same result may be arrived at more directly by integrating 
by parts the equation 

vi^ . d*u 
/* da:* 

166. We may now form the expression fot* V in terms of the 
normal co-ordinates. 

=— ^^ jmi^^i* I Wi*(ir + rn3*^*/M^'da? +...>• (1). 

If the functions u be those proper to a rod free At x = l, this expres- 
sion reduces to 



V = ^^ jm,* [u, (0?^!^ + 7/1,* [u,(l)Y<f>,' + . . . 



.......••iMf* 



In any case the equations of motion are of the form 

pcD juj^dx ^1 + — IT wii* lui'da; ^i = 4>i (3), 

and, since 4>iS<^ is by definition the work done by the impressed 
forces during the displacement S<^, 

4>i= / Yuipmdx (4), 

if Ypwdx be the lateral force acting on the element of mass ptodx. 
If there be no impressed forces, the equation reduces to 

^.+^;^^V=o (5). 

as we know it ought to do. 



268 LATERAL VIBRATIONS OF BARS. [167. 

167. The significance of the reduction of the integrals 
jv?dx to dependence on the terminal values of the function and 
its derivatives may be placed in a clearer light by the following 
line of argument. To fix the ideas, consider the case of a 
rod clamped at a? = 0, and free at x^l, vibrating in the normal 
mode expressed by u. If a small addition AZ be made to the 
rod at the free end, the form of u (considered as a function of 
x) is changed, but, in accordance with the general principle 
established in Chapter iv. (§ 88), we may calculate the period 
under the altered circumstances without allowance for the change 
of type, if we are content to neglect the square of the change. 
In consequence of tlie straightness of the rod at the place where 
the addition is made, there is no alteration in the potential 
energy, and therefore the alteration of period depends entirely 
on the variation of T. This quantity is increased in the ratio 

^dx : I v?dxy 
Jo 

whicli is also the ratio in which the square of the period is 

augmented. Now, as we shall see presently, the actual period 

varies as P, and therefore the change in the square of the period 

is in the ratio 

1 : 1 + 4A///. 

A comparison of the two ratios shews that 



Jo 



Uj^ : lu^dx = 4t : L 



The above reasoning is not insisted upon as a demonstration, 
but it serves at least to explain the reduction of which the in- 
tegral is susceptible. Other cases in which such integrals occur 
may be treated in a similar manner, but it would often require 
care to predict with certainty what amount of discontinuity in the 
varied type might be admitted without passing out of the range 
of the principle on which the argument depends. The reader 
may, if he pleases, examine the case of a string in the middle 
of which a small piece is interpolated. 

168. In treating problems relating to vibrations the usual 
course has been to determine in the first place the forms of the 
normal functions, viz. the functions representing the normal 



168.] INITIAL CONDITIONS. 269 

types, and afterwards to investigate the integral formulae by 
means of which the particular solutions may be combined to 
suit arbitrary initial circumstances. I have preferred to follow 
a different order, the better to bring out the generality of the 
method, which does iiot depend upon a knowledge of the normal 
fwnctions. In pursuance of the same plan, I shall now investigate 
the connection of the arbitrary constants with the initial circum- 
stances, and solve one or two problems analogous to those treated 
under the head of Strings. 

The general value of y may be written 

y = ^-4i cos y m^H + B^ sin ^ m^t\ u^ 



+ (ilaC08-^r/l3*^ + -BjSin-^7W,'^j Wa 

+ (1). 

so that initially 

yo = -4iUi + 4,W2+ (2), 

Kb 

yo = -7^ {wii*fiiMi + ma'Bal4a+...} (3). 

If we multiply (2) by Ur and integrate over the length of the 
rod, we get 

[y^xirdx- ArWdx (4), 

and similarly from (3) 



-Tiy^Urdx^ mr^Br\Ur^dx (5), 



formulse which determine the arbitrary constants iir, Br. 

It must be observed that we do not need to prove analytically 
the possibility of the expansion expressed by (1). If all the 
particular solutions are included, (1) necessarily represents the 
most general vibration possible, and may therefore be adapted 
to represent any admissible initial state. 

Let us now suppose that the rod is originally at rest, in its 
position of equilibrium, and is set in motion by a blow which 
imparts velocity to a small portion of it. Initially, that is, at 
the moment when the rod becomes free, y© = 0, and y© differs from 
zero only in the neighbourhood of one point {x = c). 



270 LATERAL VIBRATIONS OF BARS. [168. 

From (4) it appears that the coefficients A vanish, and from 
(5) that 

rrir^Br I Ur^dx = -^ i^r (c) I yf^dx. 

Calling fy^tprndx, the whole momentum of the blow, Y, we 
have 

£=J1Z^ ^^(g) (Q) 

and for the final solution 

„ - _^ K (£)J^) sin C^ ^*t\ + 



ltric)U.(x) . (kI) ,\ ) ,^. 



In adapting this result to the case of a rod free dX x^l, we 
may replace 



I' 



Ur^dx by \l[ur{l)\\ 

If the blow be applied at a node of one of the normal com- 
ponents, that component is missing in the resulting motion. The 
present calculation is but a particular case of the investigation 
of § 101. 

169. As another example we may take the case of a bar, 
which is initially at rest but deflected from its natural position 
by a lateral force acting at x = c. Under these circumstances 
the coefficients B vanish, and the others are given by (4), § 168. 

Now 

l.y^'^^^wl.y'd^^' 

4nd on integrating by parts 

yo-7-i »a^-yo ITS-— .^ 3-^ 






da" ^^ dx" dx daf" 



d^yo dur d'yo ,, . f ' d'.Vo ,, , 
dx' dx''da^'^^Jo~d^^ ' 



in which the terms free from the integral sign are to be taken 
between the limits; by the nature of the case y^ satisfies the 
same terminal conditions as does Ur, and thus all these terms 



169.] SPECIAL CASES. 271 

vanish at both limits. If the external force initially applied 
to the element dx be YdXy the equation of equilibrium of the 
bar gives 

P-'^'^Z'-^ (1)' 

and accordingly 

If we now suppose that the initial displacement is due to 
a force applied in the immediate neighbourhood of the point 
a? = c, we have 

and for the complete value of y at time t, 

In deriving the above expression we have not hitherto made 
any special assumptions as to the conditions at the ends, but 
if we now confine ourselves to the case of a bar which is clamped 
at d? = and free at a? = Z, we may replace 



^Ur^dx by \l[ur{l)\\ 



If we suppose further that the force to which the initial deflection 
is due acts at the end, so that c = /, we get 

When ^ = 0, this equation must represent the initial displace- 
ment. In cases of this kind a difficulty may present itself as 
to how it is possible for the series, every term of which satisfies 
the condition ^"' = 0, to represent an initial displacement in 
which this condition is violated. The fact is, that after triple 
differentiation with respect to a?, the series no longer converges 
for ic = Z, and accordingly the value of y"' is not to be anived 
at by making the differentiations first and summing the terms 
afterwards. The truth of this statement will be apparent if 
we consider a point distant dl from the end, and replace 

«'"(Z-dZ) by u"\l)-u^{l)dl, 



272 LATERAL VIBRATIONS OF BARS. [l69. 

in which u^^ (I) is equal to 

For the solution of the present problem by normal co-ordinates 
the reader is referred to § 101. 

170. The forms of the normal functions in the various par- 
ticular cases are to be obtained by determining the ratios of the 
four constants in the general solution of 

d*u __ m* 

If for the sake of brevity x' be written for (mx/l), the solution 
may be put into the form 

u=:A{cosx -h cosh a?') -hB (cos a?' -cosha?') 
-h C (sin X + sinh x) + D (sin x — sinh x) (1), 

where cosh x and sinh x are the hyperbolic cosine and sine of x, 
defined by the equations 

cosha? = J(^ + 6-*)» sinha: = J(c*-c-^) (2). 

I have followed the usual notation, though the introduction of 
a special symbol might very well be dispensed with, since 

cosh a: = cos ir, sinh ^ = — i sin to; (3), 

where i = \/(— 1) ; and then the connection between the formulae of 
circular and hyperbolic trigonometry would be moi"e apparent. The 
rules for differentiation are expressed in the equations 

-T- cosh X = sinh x, -j- sinh x = cosh x 
ax ax 

(P d' . . 

-J-. cosh X = cosh X, -J— si nh a; := sinh x, 

CM/ CM/ 

In differentiating (1) any number of times, the same four com- 
pound functions as there occur are continually reproduced. The 
only one of them which does not vanish with x is cos a?' -f cosh a?', 
whose value is then 2. 

Let us take first the case in which both ends are free. Since 
d^ulda^ and d^u/da^ vanish with x, it follows that £ = 0, i) = 0, so 
that 

u- A (cos x' + cosh a?') + (7 (sin a?' + sinh x') (4). 



:l} <H 



170. J NORMAL FUNCTIONS FOR FREE-FREE BAR. 273 

We have still to satisfy the necessary conditions when a? = i, or 
x' = m. These give 

A (- cos m + cosh m) + C (- sin m + sinh m) = 
A{ sin m + sinh m) + G(— cosm + cosh m) 

equations whose compatibility requires that 

(cosh m — cos m)* = sinh' m — sin* m, 

or in virtue of the relation 

cosh' m — sinh' m = 1 (6), 

cosm cosh wi = l (7). 

This is the equation whose roots are the admissible values of m. 
If (7) be satisfied, the two ratios of -4 : C given in (5) are equal, 
and either of them may be substituted in (4). The constant multi- 
plier being omitted, we have for the normal function 

y . . , V f True , mx) 

w = (8mm — smhm) -^cos -j- + cosh —. > 

— (cos m — cosh m) -jsin .- + sinh-y j- (8), 

or, if we prefer it, 

u = (cos m — cosh m) scos -r- + cosh —j-> 

+ (sin m+ sinh w) -jsin -y-+ sinh -j-> (9); 

and the simple harmonic component of this type is expressed by 



y=Pucosf^m'e+€) (10). 



Kb 
171. The frequency of the vibration is c>~>s^*'» ^^ which b is 

a velocity depending only on the material of which the bar is 
formed, and m is an abstract number. Hence for a given material 
and mode of vibration the frequency varies directly as k — the 
radius of gyration of the section about an axis perpendicular to the 
plane of bending — and inversely as the square of the length. These 
results might have been anticipated by the argument from dimen- 
sions, if it were considered that the frequency is necessarily 
determined by the value of I, together with that of xb — the 
only quantity depending on space, time and mass, which occurs in 

R. 18 



274 LATERAL VIBRATIONS OF BARS. [171. 

the differential equation. If everything concerning a bar be given, 
except its absolute magnitude, the frequency varies inversely as 
the linear dimension. 

These laws find an important application in the case of tuning- 
forks, whose prongs vibrate as rods, fixed at the ends where they 
join the stalk, and free at the other enda Thus the period of vibra- 
tion of forks of the same material and shape varies sjb the linear 
dimension. The period will be approximately independent of the 
thickness perpendicular to the plane of bending, but will vai'y 
inversely with the thickness in the plane of bending. When the 
thickness is given, the period is as the square of the length. 

In order to lower the pitch of a fork we may, for temporary 
purposes, load the ends of the prongs with soft wax, or file away 
the metal near the base, thereby weakening the spring. To raise 
the pitch, the ends of the prongs, which act by inertia, may be 
filed. 

The value of b attains its maximum in the case of steel, for 
which it amounts to about 5237 metres per second. For brass 
the velocity would be less in about the ratio 1*5 : 1, so that a 
tuning-fork made of brass would be about a fifth lower in pitch 
than if the material were steel. 

[For the design of steel vibrators and for rough determinations 
of frequency, especially when below the limit of hearing, the 
theoretical formula is often convenient. If the section of the bar 
be rectangular and of thickness t in the plane of vibration, k* =■ r^t-; 
and then with the above value of 6, and the values of m given 
later, we get as applicable to the gravest mode 

(clamped-free) frequency = 84590 ^/P, 

(free-free) frequency = 538400 tIP, 

I and t being expressed in centimetres. 

The first of these may be used to calculate the pitch of steel 
tuning-forka 

The lateral vibrations of a bar may be excited by a blow, as 
when a tuning-fork is struck against a pad. This method is also 
employed for the harmonicon, in which strips of metal or glass are 
supported at the nodes, in such a manner that the free vibrations 
are but little impeded. A frictional maintenance may be obtained 



171.] NORMAL FUNCTIONS FOR CLAJIPED-CLAMPED BAR. 275 

with a bow, or by the action of the wetted fingers upon a slender 
rod of glass suitably attached. The electro-magnetic maintenance 
of forks has been already considered, § 64. It may be applied with 
equal facility to the case of metal bars, or even to that of 
wooden planks carrying iron armatures, free at both ends and 
supported at the nodes. The maintenance by a stream of wind 
of the vibrations of harmonium and organ reeds may also be 
referred to. 

The sound of a bar vibrating laterally may be reinforced by a 
suitably tuned resonator, which may be placed under the middle 
portion or under one end. On this principle dinner gongs have 
been constructed, embracing one octave or more of the diatonic 

scale.] 

172. The solution for the case when both ends are clamped 
may be immediately derived from the preceding by a double dif- 
ferentiation. Since y satisfies at both ends the terminal con- 
ditions 

da* ' dx' ^' 



it is clear that y" satisfies 



dy" 



which are the conditions for a clamped end. Moreover the general 
diflferential equation is also satisfied by y". Thus we may take, 
omitting a constant multiplier, as before, 

u = (sin m — sinh m) {cos x — cosh x*] 

— (cos m — cosh m) {sin x' — sinh xf] (1), 

while m is given by the same equation as before, namely, 

cosm co8hm = l (2). 

We conclude that the component tones have the same pitch in the 
two cases. 

In each case there are four systems of points determined by 
the evanescence of y and its derivatives. Where y vanishes, there 
is a node ; where y' vanishes, a loop, or place of maximum displace- 
ment; where y" vanishes, a point of infiection; and where j/'* 
vanishes, a place of maximum curvature. Where there are in the first 
case (free-free) points of infiection and of maximum curvature, there 

18—2 



276 LATBRAL VIBRATIONS OF BARS. [172. 

are in the second (clamped-clamped) nodes and loops respectively ; 
and vice versd^ points of inflection and of maximum curvature for 
a doubly-clamped rod correspond to nodes and loops of a rod whose 
ends are free. 

173. We will now consider the vibrations of a rod clamped at 
07 = 0, and free at x^L Reverting to the general integral (1) 
§ 170, we see that A and C vanish in virtue of the conditions at 
a; = 0, so that 

w = B (cos a/ — cosh a?') + Z) (sin 0?' - sinh a?') (I). 

The remaining conditions at a? = i give 

B( cos m 4- cosh m) + D (sin m + sinh m) = | 
B (— sin m + sinh m)-\'D (cos m + cosh m)=0 ] * 

whence, omitting the constant multiplier, 



= (sin m + smh m) < cos . — cosh . - j- 
— (cos m + cosh m) jsm . — smh -.— - 



(2). 



or 



u = (cos m + cosh vi) < cos -j — cosh -j- r 

+ (sin m — sinh m)-jsin —, — sinh -^ ' (3), 

where m must be a root of 

cosm cosh m + 1=0 (4). 

The periods of the component tones in the present problem are 
thus different from, though, as we shall see presently, nearly re- 
lated to, those of a rod both whose ends are clamped, or free. 

If the value of u in (2) or (3) be differentiated twice, the re- 
sult {u") satisfies of course the fundamental differential equation. 
At a? = 0, d^vi'jda?, d^u'^jda^ vanish, but at 0? = / u" and du"/dx 
vanish. The function u' is therefore applicable to a rod clamped 
at I and free at 0, proving that the points of inflection and of 
maximum curvature in the original curve are at the same distances 
from the clamped end, as the nodes and loops respectively are 
from the fr^e end. 



174.] 



CALCULATION OF PERIODS. 



277 



174. In default of tables of the hyperbolic cosine or its loga- 
rithm, the admissible values of m may be calculated as follows. 
Taking first the equation 

cosm cosh 771 = 1 (1), 

we see that m, when large, must approximate in value to 
^(2i+l)7r, I being an integer. If we assume 

m = i(2i + l)7r-(-l)»/8 (2), 

fi will be positive and comparatively small in magnitude. 

Substituting in (1), we find 

cot J)8 = c"» = e*<*'+*>' e-^-^>^ ; 

or, if eJ<2*+i)» be called a, 

a tan i)9 = e<-^>^ (3), 

an equation which may be solved by successive approximation after 
expanding tan^)9 and g^""*)'^ in ascending powers of the small 
quantity )9. The result is 



'^ a ^ ^ a* 3a' ^ ^ 3a* 



...(4)S 



which is sufficiently accurate, even when i = 1. 

By calculation 

/3i = -0179666 - -0003228 + 0000082 - -0000002 = -0176518. 

^3) fii, 0A, 05 are found still more easily. After 0^ the first term of 
the series gives fi correctly as far as six significant figures. The 
table contains the value of )9, the angle whose circular measure is 
i8, and the value of sin ^)9, which will be required further on. 

Free-Free Bar, 



/?. 



P expressed in degrees, 
minutes, and seconds. 



1 
2 

3 ; 

4 ! 



10-^ X •176518 
10-' X -777010 
10-* X -335505 
10-» X -144989 
10-' X -626556 



P 0' 40"-94 
2' 40"-2699 
6"-92029 
•299062 
•0129237 






10-« X -88258 
10-» X -38850 
10-* X -16775 
10-* X -72494 
10-' X -31328 



^ This process is somewhat similar to that adopted by Strehlke. 



278 LATERAL VIBRATIONS OF BARS. [l74. 

The values of w which satisfy (1) are 

Wi= 4-7123890 + /3,= 47300408 
r?j,= 7-8539816 -)82= 7-8532046 
wi, = 10-9955743 + A = 10-9956078 
m, = 14-1371669 - /34 = 141371655 
m, = 17-2787596 + ^^ = 17*2787596 
after which m = ^(2i+ l)7r to seven decimal places. 

We will now consider the roots of the equation 

cosw coshm = — 1 (5)\ 

[Assuming 

r?i, = i(2i-l)7r^(-lVa, (6), 

we have e^ = cot ^Oi = g^^a'-^)' . er^-^^^i , 

or a tan ^Oj+i = g-^-^^'^+i (7), 

a having the value previously defined. 

Thus, as in (4), 

«'*.-!-(-»'^s-(-')'"^+ <■'). 

Ui+i being approarimately equal to fit. 
The values calculated from (8) are 

a, = 10-^ X -182979, a^ = IQ-* x 335527, 
a, =» 10-' X -775804, Cj = 10"* x -144989, 

after which the difference between Oj+j and )9,- does not appear.] 

The value of Ci may be obtained by trial and error from the 
equation 

logio cot ^tti - -6821882 - 43429448 a, = 0, 

and will be found to be 

a, = -3043077. 

Another method by which Wi may be obtained directly will be 
given presently. 

The values of m, which satisfy (5), are 

1)1, =» 1-5707963 + ai= 1875104 
7n,= 4-7123890-0,= 4-694098 
^8= 7-8539816 4-03= 7-854757 
m, = 10-9955743 - a^ = 10995541 
m, = 14-1371669 + a,= 14137168 
VI, = 17-2787596 - a« = 17-278759 , 

^ The calculation of the roots of (5) given in the first edition was affected by an 
error, which has been pointed out by Qreenhill (Math. Mess.^ Dec. 1S86). 



174.] CALCULATION OP PERIODS, 279 

after which m = ^ (2i — 1) tt sensibly. The frequencies are propor- 
tional to ni\ and are therefore for the higher tones nearly in the 
ratio of the squares of the odd numbers. However, in the case of 
overtones of very high order, the pitch may be slightly disturbed 
by the rotatorj' inertia, whose effect is here neglected. 

176. Since the component vibrations of a system, not subject 
to dissipation, are necessarily of the harmonic type, all the values 
of m*, which satisfy 

cosm coshm= ± 1 (1), 

must be real. We see further that, if m be a root, so are also 
— m, wV(— 1), — mV(— 1). Hence, taking first the lower sign, we 
have 

2(c..smcoshm+l)=l-j2"^12^" 



=('-3('-5)^ «• 



If we take the logarithms of both sides, expand, and equate co- 
efficients, we get 

2 — = ^; 2— ^ = ,^j.«^; &c (3). 

This is for a clamped-free rod. 

From the known value of 2m~*, the value of mi may be derived 
with the aid of approximate values of m,, m^, We find 

2m-« = 006547621, 
and mf^ = -000004 242 

7w,-» = 000000069 
mr^ = 000000005, 
whence mr^ = 006543305 

giving 7Wi = '1875104, as before. 

In like manner, if both ends of the bar be clamped or free, 

'->ll5---('-3('-S)^ <*)■ 

whence 2 — : = ts-jt^ &c., where of course the summation is exclu- 
m* 12.3o 

sive of the zero value of m. 



280 LATERAL VIBRATIONS OF BARS, [176. 

176. The frequencies of the series of tones are proportional to 
m'. The interval between any tone and the gravest of the series 
may conveniently be expressed in octaves and fractions of an 
octave. This is effected by dividing the difference of the logarithms 
of m^ by the logarithm of 2. The results are as follows : 



1-4629 


2-6478 


2-4358 


4-1332 


3-1590 


5-1036 


3-7382, &c. 


6-8288, &c 



where the first column relates to the tones of a rod both whose 
ends are clamped, or free ; and the second column to the case of a 
rod clamped at one end but free at the other. Thus fix>m the 
second column we find that the first overtone is 2*6478 octaves 
higher than the gravest tone. The fractional part may be reduced 
to mean semitones by multiplication by 12. The interval is then 
two octaves + 7*7736 mean semitones. It will be seen that the 
rise of pitch is much more rapid than in the case of strings. 

If a rod be clamped at one end aDd free at the other, the pitch 
of the gravest tone is 2 (log 4*7300 - log 1*8751) -t- log 2 or 2*6698 
octaves lower than if both ends were clamped, or both free. 

177. In order to examine more closely the curve in which the 
rod vibrates, we will transform the expression for u into a form 
more convenient for numerical calculation, taking first the case 
when both ends are free. Since m = ^(2i + l)'7r — (— 1)*/8, 
cos m = sin /8, sin m = cos iir x cos /8 ; and therefore, m being a 
root of cos m cosh m = 1, cosh m = cosec /8. 

Also 

sinh' m = cosh^ m — 1 = tan' m =• cot* /8, 

or, since cot /8 is positive, 

sinh m = cot /8. 
Thus 

sin m — sinh m __ 1 — cos iir sin 13 
cos m — cosh m cos /8 

(cos i^ — cos iir sin ^/S)* 



(cos J/8 — cos iV sin i/8)(cos ^/S + cos tV sin i/8) 

_ cos ^/8 cos tV — sin ^ y8 
"" cos ^/8 cos tV + sin i/8 ' 



177.] GRAVEST MODE FOR FREE-FREE BAR. 281 

We may therefore take, omitting the constant multiplier, 
u = (cos i/8 cos ITT 4- sin i/8) jsin -J- + sinh -j 

— (cos ^/8 cos tV - sin i/8) jcos -• - + cosh -y- >• 

= v2 cos tTT sm j-y - -J + ( - 1)* ^h 

+ sin i/S e^' - cos ITT cos^l3er^^ (1). 

If we further throw out the factor V2, and put 1=1, we 
may take 

where 

Fi = cos ITT sin [nuc — Jth + i( — l)*i8} \ 

logi^,= f/w? log e + log sin i^ — log V2 > W» 

log ± fj = - tho; log c + log cos ^/9 — log »^2 ) 

from which t^ may be calculated for different values of i and x. 

At the centre of the bar, a? = ^, and F^, F^ are numerically 
equal in virtue of e"* = cot J /8. When i is wen, these terms cancel. 
For i^i, we have i^i = (— lysin^tV, which is equal to zero when 
i is even, and to ± 1 when t is odd. When t is even, therefore, 
the sum of the three terms vanishes, and there is accordingly a 
node in the middle. 

When a? = 0, u reduces to — 2 (— 1)* sin { J 7r — i (— 1)* /8}, which 
(since /8 is always small) shews that for no value of t is there a 
node at the end. If a long bar of steel (held, for example, at the 
centre) be gently tapped with a hammer while varying points of 
its length are damped with the fingers, an unusual deadness in 
the sound will be noticed, as the end is closely approached. 

178. We will now take some particular case& 

Vibration with two nodes, i=l. 

If i = 1, the vibration is the gravest of which the rod is capable. 
Our formulae become 

i^i = - sin {x{rj^ + V 0' 40" -94) - 45* - 30' 20" '47} 

log F^ = 2054231 x + 37952391 

log /; = - 2054231 X 4- 1-8494681, 

from which is calculated the following table, giving the values of 
u for X equal to 00, 05, 10, &c. 



282 LATERAL VIBRATIONS OF BARS. [178. 

Tbe values of u : u (5) for the intermediate values of a: (in the 
last column) were found by interpolation formulae. If o, p, q, r, s, t 
be six consecutive terms, that intermediate between q and r is 

'l^'+'i^^*^-"+l|2[!+'-(r+»)l-Cp+') +»+'}• 



X 

•000 


■^1 


+ -0062408 


+■7070793 


" 


+ 1645219 


+ ■7133200 


+ 14266401 


■026 










1454176 


■050 


■5292548 


■0079059 


■5581572 


10953179 


1 ■263 134 


•075 










1-072162 


■100 


■3157243 


-0100153 


■4406006 


■7663401 


■8837528 


■125 








... 


■6969004 


■150 


+ ■0846166 


■01268T4 


■3478031 


■4451071 


■6133028 


•175 










■3341625 


■200 


- 1513020 


■0160726 


■2745503 


+ ■1394309 


+ 1607819 


■225 










- 0054711 


■250 


■3786027 


■0-203609 


■2167256 


- -1415162 


1631982 


■275 










■3109982 


•300 


■5849265 


■0257934 


■1710798 


■3880523 


■4475066 


■325 








... 


■5714137 


■350 


■7586838 


■0326753 


■1350477 


■6909608 


■6815032 


•375 








... 


■7766629 


•400 


■8902038 


■0413934 


■1066045 


■7422059 


■8559210 


■425 










■9184491 


■460 


■9721635 


■0524376 


■0841519 


■8355740 


■9635940 


■475 










■9908730 


■500 


-l^OOOOOO 


+ -0664285 


■0664283 


- ■86714.33 


- 1 -0000000 



Since the vibration curve is symmetrical with respect to the 
middle of the rod, it is unnecessary to continue the table beyond 
X - '3. The curve itself is shewn in fig. 28. 

Fig. 38. 




178.] 



FREE-FREE BAR WITH THREE NODES. 



283 



To find the position of the node, we have by interpolation 

which is the fraction of the whole length by which the node is 
distant from the nearer end. 

Vibration with three nodes, i = 2. 

i^i = sin { (45(>> - 2' 40" -27) ^r - 45« 4- 1' 20" •135} 
log F^ = 3-410604 X + 4438881 6 
log (- F;) = - 3-410604 X + 1-8494850. 



X 


u:-u{0) 


X 


1 
M : - M (0) 


•000 


- 10000 


•250 


+ -5847 


•025 


•8040 


•275 


•6374 


•050 


•6079 


•300 


•6620 


•075 


•4147 


•325 


•6569 


•100 


•2274 


•350 


•6245 


•125 


- ^0487 


•375 


•5652 


•150 


+ •ins 


•400 


•4830 


•175 


•2672 


•425 


•3805 


•200 


•3972 


•450 


•2627 


•225 


•5037 


•475 


•1340 






•500 


•0000 



In this table, as in the preceding, the values of u were calcu- 
lated directly for a? = -000, '050, -100 &c, and interpolated for the 
intermediate values. For the position of the node the table gives 
by ordinary interpolation a: =132. Calculating from the above 
formulae, we find 

w (1321) = --000076. 

w (1322) = 4-000881, 

whence a; = -132108, agreeing with the result obtained by Strehlke. 
The place of maximum excursion may be found from the derived 
function. We get 

li (-3083) = 4- -0006077, u' (3084) = - -0002227, 

whence u' (-308373) = 0. 



284 



LATERAL VIBRATIONS OF BARS. 



[178. 



Hence it is a maximum, when x = '308373 ; it then attains 
the value '6636, which, it should be observed, is much less than 
the excursion at the end. 

The curve is shewn in fig. 29. 

Fig. 29. 




Vibration tuithfour nodes, i = 3. 

i?*! = - sin { (63(y> + 6"-92) a? - 45<' - 3"-46}, 

logi?;= 4-775332 a; + 50741527, 

log F^^" 4-775332 x + 1-8494850. 

From this w(0) = 1-41424, w(i) = 1-00579. The positions of 
the nodes are readily found by trial and error. Thus 

u (-3558) = - 000037 a (3559) = + 001047, 

whence u (-355803) = 0. The value of x for the node near the end 
is -0944, (Seebeck). 

The position of the loop is best found from the derived 
function. It appears that u' = 0, when x = -2200, and then 
u = — '9349. There is also a loop at the centre, where however 
the excursion is not so great as at the two others. 

Fig. 30. 




We saw that at the centre of the bar F^ and F^ are numerically 
equal. In the neighbourhood of the middle, F^ is evidently very 
small, if t be moderately great, and thus the equation for the nodes 
reduces approximately to 



mx IT , 



-iy|= 



±ntr. 



n being an integer. If we transform the origin to the centre of 

the rod, and replace m by its approximate value ^(2i + l)7r, we 

find 

a; _ ± 2n — t 

1 ~ ~2i +T ' 



178.] GRAVEST MODE FOR A CLAMPED-FBEE BAR. 285 

shewing that near the middle of the bar the nodes are uniformly- 
spaced, the interval between consecutive nodes being 2l-7-(2i+l), 
This theoretical result has been verified by the measurements of 
Strehlke and Lissajous. 

For methods of approximation applicable to the nodes near 
the ends, when i is greater than 3, the reader is referred to the 
memoir by Seebeck already mentioned § 160, and to Donkin's 
Acoustics (p. 194). 

179. The calculations are very similar for the case of a bar 
clamped at one end and free at the other. If uo: F, and 
F=Fi'\'Fi + Fz, we have in general 

F^ = cos {mx + iir + i (- lya}, 

(-1V 1 

i^,= --^siniae»»*; i^, = --^cosia«-^. 

If i= 1, we obtain for the calculation of the gravest vibration- 
curve 

(ISO 



Fi = cos • 



IT 



maf> + 4:0^ -^8^43' '0665 



log (- Ft) = nix\oge+l -0300909. 
log (- i^s) = - Twa? log e + 1 •8444383. 



These give on calculation 
i^(0) = -000000. 
i?* (-2) = -102974, 
i^ (-4) = -370625, 



F( -6)= -743452, 
F{ -8) = 1-169632, 
i?'(10) = 1-612224, 



from which fig. 31 was constructed. 



Fig. 31. 



286 



LATERAL VIBRATIONS OF BARS. 



[179. 



The distances of the nodes from the free end in the case of a 
rod clamped at the other end are given by Seebeck and by Donkin. 

2°^ tone -2261. 



3"* tone -1321, 4999. 
4^Hone -0944, -3558, 



•6439. 



•th 



tone 



1-3222 4-9820 90007 4/- 3 4i- 10-9993 4i --70175 



4i-2' 4t-2' 4i-2'4i-2' 4i-2 



4t-2 



"The last row in this table must be understood as meaning 

47 — 3 
that j^ — ^ may be taken as the distance of the f^ node from the 

free end, except for the first three and the last two nodes." 

When both ends are free, the distances of the nodes from the 
nearer end are 



l" tone -2242. 








2™* tone -1321 


•5. 






3"* tone -0944 


•3558. 






^, , 1-3222 
* *«°^ 4i+2 


4-9820 

4i + 2 


9-0007 
4i+2 


4;- 3 

4i + 2 



The points of inflection for a free-free rod (corresponding to 
the nodes of a clamped-clamped rod) are also given by Seebeck ; — 



!•* tone 
2nd tone 
3*^ tone 

^ tone 



1** point. 



2n<l point 



rtb 



point. 



No inflection point. 



-5000 

•3593 

50175 

4i + 2 



8-9993 
4t+2 



4k + 1 
4i + 2 



Except in the case of the extreme nodes (which have no 
corresponding inflection-point), the nodes and inflection-points 
always occur in close proximity. 

180. The case where one end of a rod is free and the other 
mpported does not need an independent investigation, as it may be 



180.] POSITION OF NODES. 287 

referred to that of a rod with both ends free vibrating in an even 
mode, that is, with a node in the middle. For at the central node 
y and y" vanish, which are precisely the conditions for a supported 
end. In like manner the vibrations of a clamped-supported rod 
are the same as those of one-half of a rod both whose ends are 
clamped, vibrating with a central node. 

181. The last of the six combinations of terminal conditions 
occurs when both ends are supported. Referring to (1) § 170, we 
see that the conditions at ir = 0, give -4 = 0, -B = ; so that 

u = (C + D) sin ar' + (C - 2)) sinh x\ 

Since u and w" vanish when a?' = m, (7 — i) = 0, and sin m = 0. 

Hence the solution is 

. iirx i^ir^Kb ^ 
y = sm ^- cos— ^^-t (1), 

where i is an integer. An arbitrary constant multiplier may of 
course be prefixed, aud a constant may be added to t 

It appears that the normal curves are the same as in the case 
of a string stretched between two fixed points, but the sequence of 
tone is altogether diflferent, the frequency varying as the square 
of i. The nodes and inflection-points coincide, and the loops 
(which are also the points of maximum curvature) bisect the 
distances between the nodes. 

182. The theory of a vibrating rod may be applied to illustrate 
the general principle that the natural periods of a system fulfil the 
maximum-minimum condition, and that the greatest of the natural 
periods exceeds any that can be obtained by a variation of 
type. Suppose that the vibration curve of a clamped-free rod is 
that in which the rod would dispose itself if deflected by a force 
applied at its free extremity. The equation of the curve may be 
taken to be 

y = -91x^ + 0^, 

which satisfies d^yjdx^ = throughout, and makes y and y vanish 
at 0, and y" at L Thus, if the configuration of the rod at time t be 

y = (-3ii;» + .T») QOBpt (1). 

the potential energy is by (1) § 161, 6 j/c'wZ'cos'p^, while the 



288 LATERAL VIBRATIONS OF BARS. [182. 

kinetic energy is =^ pto U p^ sm' pt ; and thus jp* = -- 
Now j>i (the true value of p for the gravest tone) is equal to 



^ X {ismr ; 



80 that 



p,:p^ (1-8751)» y ^ = -98556, 



shewing that the real pitch of the gravest tone is rather (but 
comparatively little) lower than that calculated from the hypo- 
thetical type. It is to be observed that the h)rpothetical type in 
question violates the terminal condition y'"= 0. This circumstance, 
however, does not interfere with the application of the principle, 
for the assumed type may be any which would be admissible as an 
initial configuration ; but it tends to prevent a very close agree- 
ment of periods. 

We may expect a better approximation, if we found our calcu- 
lation on the curve in which the rod would be deflected by a force 
acting at some little distance from the free end, between which 
and the point of action of the force (a? = c) the rod would be 
straight, and therefore without potential energy. Thus 

potential energy = 6 qx^axs^ cos^ pt. 

The kinetic energy can be readily found by integration from 
the value of y. 

From to c y = — 3ca7* 4- a?* ; 

and from ctol y^(f{C'' Sx), 

as may be seen frx)m the consideration that y and y' must not 
suddenly change at a? = c. The result is 

[33 1 

=7xC' + ic*(Z — c)(c*+3P) , 

whence 

The maximum value of 1/p^ will occur when the point of 
application of the force is in the neighbourhood of the node of the 
second normal component vibration. If we take c = f Z, we obtain 
a result which is too high in the musical scale by the interval 



182.] LOADED END. 289 

expressed by the ratio 1 : '9977, and is accordingly extremely near 
the truth. This example may give an idea how nearly the period 
of a vibrating system may be calculated by simple means without 
the solution of differential or transcendental equations. 

The type of vibration just considered would be that actually 
assumed by a bar which is itself devoid of inertia, but carries a 
load M at its free end, provided that the rotatory inertia of ilf could 
be neglected. We should have, in fact, 

F= QqK^toV cos-' pU T = ^MU'p^hm^pt, 

so that P'--M^ (3). 

Even if the inertia of the bar be not altogether negligible in 
comparison with if, we may still take the same type as the basis 
of an approximate calculation : 

F=6?/c='g)Pco8«j^«, 

T = UmI^ + 1? ptoV\ p^ sin' pt, 
whence 



p' ~ 3gr/e=^G) \ 



^+i1)^«0 ^*>' 



that is, M is to be increased by about one quarter of the mass of 
the rod. Since this result is accurate when M is infinite, and does 
not differ much from the truth, even when il/ = 0, it may be re- 
garded as generally applicable as an approximation. The error 
will always be on the side of estimating the pitch too high. 

183. But the neglect of the rotatory inertia of M could not 
be justified under the ordinary conditions of experiment. It is as 
easy to imagine, though not to construct, a case in which the inertia 
of translation should be negligible in comparison with the inertia of 
rotation, as the opposite extreme which has just been considered. 
If both kinds of inertia in the mass M be included, even though 
that of the bar be neglected altogether, the system possesses two 
distinct and independent periods of vibration. 

Let z and 6 denote the values of y and dy/dx B,t x^l. Then 
the equation of the curve of the bar is 

y=-p-^ + — ^^, 
R. 19 



290 
and 



LATERAL VIBRATIONS OF BARS. 



[183. 



y^^^iSz-'-SzW + l^e^} (1); 



while for the kinetic energy 

T^iMz' + ^MK'^e' (2), 

if K be the radius of gyration of M about an axis perpendicular to 
the plane of vibration. 

The equations of motion are therefore 

Mz +^^ (6z-SW) = o' 

Mk''0-^^^(^SIz + 21'0) = 
whence, if z and vary as cos pt, we find 



(3); 



P 



(O 






1 + 



3«'» / 3/c'» 9*'*1 



(4). 



corresponding to the two periods, which are always different. 

If we neglect the rotatory inertia by putting ic' = 0, we fall 
back on our previous result 

P " "MP • 

The other value of p* is then infinite. 

It K : I be merely small, so that its higher powers may be 
neglected, 

P MIk'' \ ^i I*) 



(5). 



If on the other hand «'* be very great, so that rotation is 
prevented. 






or 



MU^ 



(6), 



the latter of which is very small. It appears that when rotation 
is prevented, the pitch is an octave higher than if there were no 
rotatory inertia at all. These conclusions might also be derived 



183.] EFFECT OF ADDITIONS. 291 

directly from the differential equations ; for if /c' = « , tf = 0, and 
then 

but if #c' = 0, = 3z/2l, by the second of equations (3), and in 
that case 

M£ + -^ — 2r = 0. 

184. If any addition to a bar be made at the end, the period 
of vibration is prolonged. If the end in question be free, suppose 
first that the piece added is without inertia. Since there would be 
no alteration in either the potential or kinetic energies, the pitch 
would be unchanged; but in proportion as the additional part 
acquires inertia, the pitch falls (§^88). 

In the same way a small continuation of a bar beyond a 
clamped end would be without effect, as it would acquire no 
motion. No chlmge will ensue if the new end be also clamped ; 
but as the first clamping is relaxed, the pitch falls, in consequence 
of the diminution in the potential energy of a given deformation. 

The case of a ' supported ' end is not quite so simple. Let the 
original end df the rod be A, and let the added piece which is at 
first supposed to have no inertia, he AB. Initially the end A is 
fixed, or held, if we like so to regard it, by a spring of infinite stiff- 
ness. Suppose that this spring, which has no inertia, is gradually 
relaxed. During this process the motion of the new end B 
diminishes, and at a certain point of relaxation, B comes to rest. 
During this process the pitch falls. B, being now at rest, may be 
supposed to become fixed, and the abolition of the spring at A 
entails another fall of pitch, to be further increased as AB acquires 
inertia. 

186. The* case of a rod which is not quite uniform may be 
treated by the general method of § 90. We have in the notation 
there adopted 



a^ = \ pfO(,u-f*dx, BOf — I BpctUf^dx, 



19—2 



292 LATERAL VIBRATIONS OF BARS. [185. 

whence, P^ being the uncorrected value of jo^, 

Pr " ■* r "(•'•'• 



\^^^-£)^ jp^oUr^^ 



*■ \ B^jUr^dx payjur^dx) 

[If the motion be strictly periodic with respect to a*, w/' is 
proportional to tir, and both quantities vanish at a node. Ac- 
cordingly an irregularity situated at a node of this kind of motion 
has no effect upon the period. A similar conclusion will hold good 
approximately for the interior nodes of a bar vibrating with 
numerous subdivisions, even though, as when the terminals are 
clamped or free, the mode of motion be not strictly periodic with 
respect to w.] 

If the rod be clamped at and free at I, 



p^u — . _____ - 



Jo ^0 fr^r J DG)« ) 



i_ 

The same formula applies to a doubly free bar. 
The effect of a small load dM is thus given by 

B^^L^.u^dM) 

where M denotes the mass of the whole bar. If the load be at 
the end, its effect is the same as a lengthening of the bar in the 
ratio M : M + dM. (Compare § 167.) 

[In (2) dM is supposed to act by inertia only ; but a similar 
formula may conveniently be employed when an irregularity of 
mass dM depends upon a variation of section, without a change 
of mechanical properties. Since B = q/c*a>, 

so that the effect of a local excrescence is given by 

If the thickneds in the plane of bending be constant, Sk' = 0, 
and S (/e*a>)/(/ic'6>), = Sa/a^ 



185.] CORRECTION FOR ROTATORY INERTIA. 293 

t:, ^, fSooda: dM 

^''''^''' J 7^ = ^ ' 

and thus p./p. = 1 + 4 ^ '^^^ (4). 

If, however, the thickness in the plane perpendicular to that of 
bending be constant, and in the plane of bending variable (27), 

then S (/e*a))/(ic«o))o = V/7o* = 3 S7/70 = 3Sa)/a)o ; 

and in place of (4) 

pVP^^i-^^-j^-^^ (^)- 

If a tuning-fork be filed {dM negative) near the stalk (clamped 
end), the pitch is lowered ; and if it be filed near the free end, the 
pitch is raised. Since u^*^ = uf, the effects of a given stroke of 
the file are equal and opposite in the circumstances of (4), but in 
the circumstances of (5) the effect at the stalk is three times as 
great as at the free end.] 

186. The same principle may be applied to estimate the 
correction due to the rotatory inertia of a uniform rod. We have 
only to find what addition to make to the kinetic energy, supposing 
that the bar vibrates according to the same law as would obtain, 
were there no rotatory inertia. 

Let us take, for example, the case of a bar clamped at and 
free at Z, and assume that the vibration is of the type, 

where u is one of the functions investigated in § 179. The kinetic 
energy of the rotation is 

= ^'"*'*"^ 8in°ja< (2W + »»«'«),, 
by (2) § 165. 

To this must be added 

^/>'sin'/}M u^dx, or -^ jo^sin'p^V; 

I J Q o 

so that the kinetic energy is increased in the ratio 

1 : 1+ y,- (2- + m— • 



294 LATERAL VIBRATIONS OF BARS. [l86. 

The altered frequency bears to that calculated without allow- 
ance for rotatory inertia a ratio which is the square root of the 
reciprocal of the preceding. Thus 






By use of the relations cosh m = — sec m, sinh m = cos tV. tan m, 
we may express u' : u when x^l'in the form 



u' — sin m cos a 



u cos iir + cos m 1 — cos iir sin a ' 
if we substitute for m from 

m = i(2i-l)^-(-iya. 

In the case of the gravest tone, a = 3043, or, in degrees and 
minutes, a = 17® 26', whence 

- = -73413, 2--^m — = 2-4789. 
u u vJ^ 

Thus 

p:P=l-2-324l| (2), 

which gives the correction for rotatory inertia in the case of the 
gravest tone. 

When the order of the tone is moderate,»a is very small, 

and then 

u' : w = 1 sensibly, 



and 






shewing that the correction increases in importance with the 
order of the component. 

In all ordinary bars k \lia very small, and the term depending 
on its square may be neglected without sensible error. 

187. When the rigidity and density of a bar are variable 
from point to point along it, the normal functions cannot in 
general be expressed analytically, but their nature may be investi- 
gated by the methods of Sturm and Liouville explained in § 142. 

If , as in § 162, B denote the variable ilexural rigidity at any 



187.] ROOTS OF COMPOUND FUNCTIONS. 295 

point of the bar, and pto dx the mass of the element, whose length 
is dx, we find as the general differential equation 

^^^p^'2-' w. 



da?\ da?) 



the effects of rotatory inertia being omitted. If we assume that 
y <x cos vt, we obtain as the equation to determine the form of the 
normal functions 



£(^g)-« «• 



in which v^ is limited by the terminal conditions to be one of an 
infinite series of definite quantities v^, v^^ v^ 

Let U8 suppose, for example, that the bar is clamped at both 
ends, so that the terminal values of y and dyjdx vanish. The first 
normal function, for which v^ has its lowest value vi, has no 
internal root, so that the vibration-curve lies entirely on one side 
of the equilibrium-position. The second normal function has one 
internal root, the third function has two internal roots, and, 
generally, the r^** function has r — 1 internal roots. 

Any two different normal functions are conjugate, that is to 
say, their product will vanish when multiplied by ptadx, and 
integrated over the length of the bar. 

Let us examine the number of roots of a function f(x) of 
the form 

compounded of a finite number of normal functions, of which the 
function of lowest order is Um(^) and that of highest order is 
Wn i^)' If the number of internal roots oif{x) be m, so that there 
are /i + 4 roots in all, the derived function f (x) cannot have less 
than /x -h 1 internal roots besides two roots at the extremities, and 
the second derived function cannot have less than /x + 2 roots. 
No roots can be lost when the latter function is multiplied by B, 
and another double differentiation with respect to x will leave at 
least fM internal roots. Hence by (2) and (3) we conclude that 

has at least as many roots as f{x). Since (4) is a function of the 
same form as f{x\ the same argument may be repeated, and a 
series of functions obtained, every member of which has at least 



296 LATERAL VIBRATIONS OF BARS. [187. 

as many roots as f{x) has. When the operation by which (4) was 
derived from (3) has been repeated sufficiently often, a function is 
arrived at whose form differs as little as we please from that of the 
component normal function of highest order %in{x)\ and we con- 
clude that f{x) cannot have more than n — 1 internal roots. In 
like manner we may prove that f{x) cannot have less than m — 1 
internal roots. 

The application of this theorem to demonstrate the possibility 
of expanding an arbitrary function in an infinite series of normal 
functions would proceed exactly as in § 142. 

[An analytical investigation of certain cases where the section 
of a rod is supposed to be variable, will be found in a memoir by 
Kirchhoff^]. 

188. When the bar, whose lateral vibrations are to be con- 
sidered, is subject to longitudinal tension, the potential energ}' of 
any configuration is composed of two parts, the first depending on 
the stiffness by which the bending is directly opposed, and the 
second on the reaction against the extension, which is a necessary 
accompaniment of the bending, when the ends are nodes. The 
second part is similar to the potential energy of a deflected string ; 
the first is of the same nature as that with which we have been 
occupied hitherto in this Chapter, though it is not entirely 
independent of the permanent tension. 

Consider the extension of a filament of the bar of section dw, 
whose distance from the axis projected on the plane of vibration 
is i;. Since the sections, which were normal to the axis originally, 
remain normal during the bending, the length of the filament 
bears to the corresponding element of the axis the i*atio R + ri :R, 
R being the radius of curvature. Now the axis itself is extended 
in the ratio q'*q-¥ T^ reckoning from the unstretched state, if 
Tm denote the whole tension to which the bar is subjected. 
Hence the actual tension on the filament is [T+ri{T •\-q)IR]dtD^ 
from which we find for the moment of the couple acting across the 
section 

1 Berlin MonaUber,, 1S79 ; Collected Works, p. 339. See also Todhunter and 
Pearson's Hiitory of the Theory of Elasticity, Vol. ii., Part ii., § 1302. 



188.] PERMANENT TENSION. 297 

and for the whole potential energy due to stiffness 

Uq+T)K'a>f[^Jda: (1), 

an expression differing from that previously used (§ 162) by the 
substitution of (q + T) for q. 

Since q is the tension required to stretch a bar of unit area to 
twice its natural length, it is evident that in most practical cases 
T would be negligible in comparison with q. 

The expression (1) denotes the work that would be gained 
during the straightening of the bar, if the length of each element 
of the axis were preserved constant during the process. But 
when a stretched bar or string is allowed to pass from a displaced 
to the natural position, the length of the axis is decreased. The 
amount of the decrease is if(dy/dwyd^, and the corresponding 
gain of work is 



Thus 



V^Hq + T)^a>j{^Jdx + ^Ta>f[^Jda: (2). 

The variation of the first part due to a hypothetical displace- 
ment is given in § 162. For the second part, we have 

In all the cases that we have to consider, Sy vanishes at the 
limits. The general differential equation is accordingly 

or, if we put q-\-T=b^p, T= a«p, 

'^V da^ da^dt^l ""da^^dt^^ ^*^* 

For a more detailed investigation of this equation the reader is 
referred to the writings of Clebsch^ and Donkin. 

189. If the ends of the rod, or wire, be clamped, dy/dx = 0, and 
the terminal conditions are satisfied. If the nature of the support 
be such that, while the extremity is constrained to be a node, there 

1 Theorie der ElaiticitUt fester lOVrper. Leipzig, 1862. 



298 LATERAL VIBRATIONS OF BARS. [189. 

is no couple acting on the bar, d^y/da^ must vanish, that is to say, 
the end must be straight. This supposition is usually taken to 
represent the case of a string stretched over bridges, as in many 
musical instruments ; but it is evident that the part beyond the 
bridge must partake of the vibration, and that therefore its length 
cannot be altogether a matter of indiflference. 

If in the general differential equation we take y proportional 
to cos ntt we get 

-('■S-'2)--2-"V-o a). 

which is evidently satisfied by 

y = sin i , cos nt (2), 

if n be suitably determined. The same solution also makes 
y and y" vanish at the extremities. By substitution we obtain 
for n, 

"^ " P > + iV/c^ ^^^' 

which determines the frequency. 

If we suppose the wire infinitely thin, n' = i'7r*a'-r P, the same 
as was found in Chapter vi., by starting from the supposition of 
perfect flexibility. If we treat k:1 as b, very small quantity, the 
approximate value of n is 



21' Va' J, 



I { ' 21 

For a wire of circular section of radius r, /k* = J r*. and if we 
replace b and a by their values in terms of q, T, and p, 



lira 






which gives the correction for rigidity \ Since the expression 
within brackets involves i, it appears that the harmonic relation 
of the component tones is disturbed by the stiffness. 

190. The investigation of the correction for stiffness when the 
ends of the wire are clamped is not so simple, in consequence of 
the change of type which occurs near the ends. In order to pass 
from the case of the preceding section to that now under con- 

1 Donkin*8 Acoutties, Art. 1S4. 



190.] 



PERMANENT TENSION. 



299 



sideration an additional constraint must be introduced, with the 
effect of still further raising the pitch. The following is, in the 
main, the investigation of Seebeck and Donkin. 

If the rotatory inertia be neglected, the differential equation 
becomes 



(^-;^*-;y!'=» <■)■ 



where D stands for -y- . In the equation 



a? 



n' 



K^b^ b'K^ 



= 0, 



one of the values of D* must be positive, and the other negative. 
We may therefore take 



^-^.^-i.S=<^-'><^+'^> 



(2), 



and for the complete integral of (1) 

y — A cosh ax-\-B sinh ax-\-Ccosfix-\- D sin fix (3), 

where a and fi are functions of n determined by (2). 

The solution must now be made to satisfy the four boundary 
conditions, which, as there are only three disposable ratios, lead 
to an equation connecting a, )8, L This may be put into the form 

sinhaZ sinfil 2a fi 



1 — cosh al cos fil a" — /8^ 



= 



.(4). 



a" 



The value of -j—^oi » determined by (2), is — :^- , so that 

sinh al sin fit 
1 — cosh al cos (il 

From (2) we find also that 






(5). 



^ 



= «-l / 
2fr»«» (V 



1 + 4 — . +1 



tt' 



1 + 4 - ^ 1 

a* 



.(6). 



Thus far our equations are rigorous, or rather as rigorous as 
the differential equation on which they are founded ; but we shall 
now introduce the supposition that the vibration considered is but 



300 LATERAL VIBRATIONS OF BARS. [190. 

slightly affected by the existence of rigidity. This being the case, 
the approximate expression for y is 



y = sm -^- cos ( y c^H > 



%Trx 
I 
and therefore 

fi = iw/lf n^iirajl (7), 

nearly. 



The introduction of these values into the second of equations 
(6) proves that n^b^K^/a* or h^K^ja^l^ is a small quantity under the 
circumstances contemplated, and therefore that a^Z^ is a large 
quantity. Since coshaZ, sinhaZ are both large, equation (5) re- 
duces to 

tan/8Z=:- /, 

or, on substitution of the approximate value for /8 derived fi'om 

(6), 

nl ^ nbx 
tan — = 2 - . 
a a* 

The approximate value of nl/a is iV. If we take rd/a = iir + 0, 
we get 

tan(t7r + tf) = tantf=tf = 2-^f = 2t7r-y, 

so that n = i-^(l + 2^j) (8). 

According to this equation the component tones are all raised 
in pitch by the same small interval, and therefore the harmonic 
relation is not disturbed by the rigidity. It would probably be 
otherwise if terms involving m^ : P were retained ; it does not there- 
fore follow that the harmonic relation is better preserved in spite 
of rigidity when the ends are clamped than when they are free, 
but only that there is no additional disturbance in the former 
case, though the absolute alteration of pitch is much greater. It 
should be remarked that 6 : a or \/(9 + T) : '^T, is a large quantity, 
and that, if our result is to be correct, k : I must be small enough 
to bear multiplication by 6 : a and yet remain small. 

The theoretical result embodied in (8) has been compared with 
experiment by Seebeck, who found a satisfactory agreement. The 
constant of stifiness was deduced from observations of the rapidity 



190.J RESULTANT OF TWO TRAINS OF WAVES. 301 

of the vibratious of a small piece of the wire, when one end was 
clamped in a vice. 

[As the result of a second approximation Seebeck gives 
{loc. cit) 

„ = ^,{l + 4^J + (12 + »'^)^^'} (9)]. 



191. It has been shewn in this chapter that the theory of bars, 
even when simplified to the utmost by the omission of unimportant 
quantities, is decidedly more complicated than that of perfectly 
flexible strings. The reason of the extreme simplicity of the 
vibrations of strings is to be found in the fact that waves of the 
harmonic type are propagated with a velocity independent of the 
wave length, so that an arbitrary wave is allowed to travel without 
decomposition. But when we pass from strings to bars, the con- 
stant in the dififerential equation, viz. ch//dt*-\- K^b*d*j//da!^ = 0, is 
no longer expressible as a velocity, and therefore the velocity of 
transmission of a train of harmonic waves cannot depend on the 
differential equation alone, but must vary with the wave length. 
Indeed, if it be admitted that the train of harmonic waves can 
be propagated at all, this consideration is sufficient by itself to 
prove that the velocity must vary inversely as the wave length. 
The same thing may be seen from the solution applicable to 

27r 
waves propagated in one direction, viz. y = cos -r-iVt — x\ which 

satisfies the differential equation if 

VJ-^ (1). 

Let us suppose that there are two trains of waves of equal 
amplitudes, but of different wave lengths, travelling in the same 
direction. Thus 



= 2 COS TT 



l'(;-;')-'a4)l-'|'(;-?)-'(^x^)!-«- 

If t' — T, X' — X be small, we have a train of waves, whose 
amplitude slowly varies from one point to another between the 
values and 2, forming a series of groups separated from one 
another by regions comparatively free from disturbance. In the 
case of a string or of a column of air, X varies as r, and then the 



302 LATERAL VIBRATIONS OF BARS. [l91. 

groups move forward with the same velocity as the component trains, 
and there is no change of type. It is otherwise when, as in the case 
of a bar vibrating transversely, the velocity of propagation is a 
function of the wave length. The position at time t of the middle 
of the group which was initially at the origin is given by 

which shews that the velocity of the group is 

If we suppose that the velocity F of a train of waves varies as 
X**, we find 

^-^=im*-<»->^ <»>• 

In the present case n = — 1, and accordingly the velocity of the 
groups is ttvice that of the component waves^ 

192. On account of the dependence of the velocity of propaga- 
tion on the wave length, the condition of an infinite bar at any 
time subsequent to an initial disturbance confined to a limited 
portion, will have none of the simplicity which characterises the 
corresponding problem for a string; but nevertheless Fourier's 
investigation of this problem may properly find a place here. 

It is required to determine a function of x and t, so as to 
satisfy 

^y+^ = o (1) 

and make initially y = ^ (x), y = ylr (x). 
A solution of (1) is 

y = cos qH cos g (a? — a) (2), 

where q and a are constants, from which we conclude that 

y = I da F{a) I dq cos qH cos g (a? — a) 

^ In the oorresponding problem for waves on the earfaoe of deep water, the 
Telocity of propagation varies directly as the square root of the wave length, so 
that n a J . The velocity of a group of such waves is therefore one half of that of 
the component trains. [See note on Progressive Waves, appended to this volume.] 



192.] Fourier's solution. 303 

is also a solution, where F{d) is an arbitrary function of a. If 
now we put t = 0, 

r+oo r+flo 

yo = I daF{a) I dq cos g (a? — a), 

which shews that F(a) must be taken to be ^- ^ (a), for then by 

/ LIT 

Fourier's double integral theorem y© = <^ (^)- Moreover, y = ; 
hence 

y = ^l da<f>(a)l dqcosq^t coaq(x — a) (3) 

satisfies the differential equation, and makes initially 

By Stokes' theorem (§ 95), or independently, we may now 
supply the remaining part of the solution, which has to satisfy the 
differential equation while it makes initially y = 0, y = -^ (a?) ; it is 

y^^-^l dayjr(a)\ dq-sinqH C09q(x — a) (4). 

27r j _oo . — ao q 

The final result is obtained by adding the right-hand members 
of (3) and (4). 

In (3) the integration with respect to q may be effected by 
means of the formula 

j^ dqcosqHcosqz = j^'^ sin(| + y (5), 

which may be proved as follows. If in the well-known integral 
formula 



J _« a 



we put iT + 6 for x, we get 



r* g-a« (*»+>&*) ^ = V? ga«6«^ 



— 00 



Now suppose that a' = i = e*'', where i = \/(— l)i and retain 
only the real part of the equation. Thus 

I COS (x^ + 26a?) dx = V^ sin (6^ -f i tt), 

^30 



304 LATERAL VIBRATIONS OF BARS. [l92. 

whence 

/+« _ 

cos a? cos 26a? dx = Jnr sin (6* + 1 w), 

from which (5) follows by a simple change of variable. Thus 
equation (3) may be written 



.« a — a? 



1 r+« 



1 /•+« 
y = -^ I d/x(co8 /!» + sin yif) <f>(x + 2/x VO (6). 

192 a. If the axis of the rod be curved instead of straight, 
we obtain problems which may be regarded as extensions of 
those of the present and of the last chapters. The most impor- 
tant case under this head is that of a circular ring, whose section 
we will regard as also circular, and of radius (c) small in 
comparison with the radius (a) of the circular axis. 

The investigation of the flexural modes of vibration, executed 
in the plane of the ring, is analogous to the case of a cylinder 
(see § 233), and was first efifected by Hopped If s be the number 
of periods in the circumference, the coefficient p of the time in 
the expression for the vibrations is given by 

^ 4 l + «' pa* ^ ^' 

where q is Young's modulus and p the density of the material. 
This may be compared with equation (9) § 233. To fall back 
upon the case of a straight axis we have only to suppose 
8 and a to be infinite in such a manner that 27ra/8 is equal to the 
proposed linear period. The vibrations in question are then purely 
transverse. 

In the class of vibrations considered above the circular axis 
remains unextended, and (§ 232) the periods are comparatively 
long. For the other class of vibrations in the plane of the ring, 
Hoppe found 

;>'=(i+«')^^, (2). 

1 CretU, Bd. 63, p. 168, 1871. 



192 a.] CIRCULAR RINGS. 305 

The frequencies are here independent of c, and the vibrations 
are analogous to the longitudinal vibrations of straight rods. 

If 8 = in (2), we have the solution for vibrations which are 
purely radial. 

For flexural vibrations perpendicular to the plane of the 
ring, the result^ corresponding to (1) is 

the difference consisting only in the occurrence of Poisson's ratio 
(ji) in the denominator. 

Our limits will not allow of our dwelling further upon the 
problem of this section. A complete investigation will be found 
in Love's Treatise on Elasticity, Chapter xviii. The effect of 
a small curvature upon the lateral vibrations of a limited bar 
has been especially considered by Lamb*. 

^ MicheU, Meuenger of Mathematict, xn., 1889. 
3 Proc. Lond. Math, Soc., xiz., p. 366, 1888. 



20 



CHAPTER IX. 



VIBRATIONS OF MEMBRANES. 



193. The theoretical membrane is a perfectly flexible and 
infinitely thin lamina of solid matter, of uniform material and 
thickness, which is stretched in all directions by a tension so great 
as to remain sensibly unaltered during the vibrations and displace- 
ments contemplated. If an imaginary lii)e be drawn across the 
membrane in any direction, the mutual action between ^he two 
portions separated by an element of the line is proportional to the 
length of the element and perpendicular to its direction. If the 
force in question be Ti ds, Z\ may be called the tension of the mem^ 
brane ; it is a quantity of one dimension in mass and — 2 in time. 

The principal problem in connection with this subject is the 
investigation of the transverse vibrations of membranes of different 
shapes, whose boundaries are fixed. Other questions indeed may 
be proposed, but they are of comparatively little interest ; and, 
moreover, the methods proper for solving them will be sufficiently 
illustrated in other parts of this work. We may therefore proceed 
at once to the consideration of a membrane stretched over the 
area include() within a fixed, closed, plane boundary. 

194. Taking the plane of the boundary as that of ocy, let w 
denote the small displacement therefrom of any point P of the 
membrane. Round P take a small area £>, and consider the forces 
acting upon it parallel to z. The resolved part of the tension is 
expressed by 

where ds denotes an element of the boundary of S, and dn an 
element of the normal to the curve drawn outwards. This is 
balanced by the reaction against acceleration measured by p8w^ 



194.] EQUATION OF MOTION* 307 

p being a symbol of one dimension in mass and —2 in length 
denoting the superficial density. Now by Green's theorem, if 

[f^ds^jl V^w dS = V'w . 8 ultimately, 

and thus the equation of motion is 

dJ^w ^ Ti fd?w d?w\ 

cft» " p Vda:* df) ^^' 

The condition to be satisfied at the boundary is of course w^O. 

The differential equation may also be investigated from the 
expression for the potential energy, which is found by multiplying 
the tension by the superficial stretching. The altered area is 

and thus 

^-i-.//{(£r-Q*}«^ «■ 

from which SFis easily found by an integration by parts. 

If we write Ti -r p = c*, then c is of the nature of a velocity, and 
the differential equation is * 

d^w « fd^w d?w 



=»-(S-S) w 



dV" \da^^ df 

196. We shall now suppose that the boundary of the mem- 
brane is the rectangle formed by the coordinate axes and the lines 
x = a, y = b. For every point within the area (3) § 194 is satisfied, 
and for every point on the boundary m; = 0. 

A particular integral is evidently 

w = 8m ain—j^coapt (1), 

where p. = c',r'(jV|) (2), 

and m and n are integers; and from this the general solution may 
be derived. Thus 



tn—oo n— 00 



^ = 2^j \^^ sm -^ sm -^ [A„,n cos pt + B^n smp^} (3). 

20—2 



308 VIBRATIONS OF MEMBRANES. [195. 

That this result is really general may be proved a posteriori, 
by shewing that it may be adapted to express arbitrary initial 
circumstances. 

Whatever function of the co-ordinates w may be, it can be ex- 
pressed for all values of x between the limits and a by the series 

Xism --h FjSm h , 

a a 



where the coefficients Fi, F,, &c. are independent of x. Again 
whatever function of y any one of the coefficients F may be, it can 
be expanded between and b in the series 

(7isin^+ (7jsin^ + , 





where Ci &c. are constants. From this we conclude that any 
function of x and y can be expressed within the limits of the rect- 
angle by the double series 



m-flo n-oo ^^^ ^^y 

2 2 -Awnsm sm— r^; 



m-l n-1 a 



and therefore that the expression for w in (3) can be adapted to 
arbitrary initial values of w and w. In fact 

The character of the normal functions of a given rectangle, 

. Tmwx . niry 

sm sm j- , 

a b 

as depending on m and n, is easily understood. If m and n be both 
unity, w retains the same sign over the whole of the rectangle, 
vanishing at the edge only; but in any other case there are 
nodal lines running parallel to the axes of coordinates. The 
number of the nodal lines parallel to x is n — 1, their equations 
being 

^ n* n ' n 



195.] RECTANGULAR BOUNDARY. 309 

In the same way the equations of the nodal lines parallel to y 



are 

_a 2a (m— l)a 

m VI m 

being m — 1 in number. The nodal system divides the rectangle 
into mn equal parts, in each of which the numerical value of k; is 
repeated 

196. The expression for w in terms of the normal functions 
is 

w = Z2d>,„n8m sm ,-- (1), 

a 

where ^,nn &c. are the normal coordinates. We proceed to form 
the expression for V in terms of <l>mn- We have 

In integrating these expressions over the area of the rectangle 
the products of the normal coordinates disappear, and we find 



"•I/IKS'- (D'H* 



r,a67r> 
2 4 



2SS' + 9*»»' <2). 



the summation being extended to all integral values of m and n. 

The expression for the kinetic energy is proved in the same 
way to be 

r=|^22^„' (3), 

from which we deduce as the normal equation of motion 

In this equation 

^mn^j J Zsin^—^ sin^ dxdy (5), 

a Zdxdy denote the transverse force acting on the element dxdy. 



310 VIBRATIONS OF MEMBRANES. [196. 

Let us suppose that the initial condition is one of rest under 
the operation of a constant force Zy such as may be supposed to 
arise from gaseous pressure. At the time f = 0, the impressed 
force is removed, and the membrane left to itself Initially the 
equation of equilibrium is 

^ + j,j(^). = ^*».» (6). 

whence (^wn)© is to be found. The position of the system at time t 
is then given by 

^n = (<Amn)o cos ^^^ + ^ . CTrfj (7), 

in conjunction with (1). 

In order to express 4>^n, we have merely to substitute for Z its 
value in (5), or in this case simply to remove Z from under the 
integral sign. Thus 

^mn—Z\ I sm sm-r^dady, 

JoJo a *^ 



nh 

= Z r (1 — COS rnir) (1 — cos nir). 



We conclude that <E>mn vanishes, unless m and n are hoth odd, and 
that then 

mnTT* 






Accordingly, m and n being both odd, 

_ 16Z cospt 
*"*'*^7^'^^« ^^^' 

where ^.^c^^sg+gj (9). 

This is an example of (8), § 101. 

If the membrane, previously at rest in its position of equili- 
brium, be set in motion by a blow applied at the point (a, /3), the 
solution is 

<f>mn = -J- sm sin — ,- 1 1 WqO^ dy.smpt... (10). 

[As an example of forced vibrations, suppose that a harmonic 
force acts at the centre. Unless m and n are both odd, <[>mn = 0, 
and in the case reserved 

*mn=±^iC08 9^ (11), 



196.] RECTANGULAR BOUNDARY. 311 

where Z^ is the whole force acting at time t, and ± represents 
sin^mTT sin^WTT. From (4) and (9) we have 



_ ±4ZxC03g^ 



and w is then given by (1). 

In the case of a square membrane, j) is a symmetrical function 
of m and n. When m and n are unequal, the terms occur in pairs, 
such as 

± 4Z^9^ Jsin^'^^sin ^ + sin^-^sin ^...(13), 

a combination symmetrical as between x and ^. The vibration is 
of course similarly related as well to the four sides as to the four 
comers of the square. 

In the neighbourhood of the centre, where the force is applied, 
the series loses its convergency, and the displacement w tends to 
become (logarithmically) infinite.] 

197. The frequency of the natural vibrations is found by 
ascribing different integral values to m and n in the expression 



h-y%*t w 



For a given mode of vibration the pitch falls when either 
side of the rectangle is increased. In the case of the gravest 
mode, when m = l, n = l, additions to the shorter side are the 
more effective; and when the form is very elongated, additions 
to the longer side are almost without eflfect. 

When a' and 6' are incommensurable, no two pairs of values 
of m and n can give the same frequency, and each fundamental 
mode of vibration has its own characteristic period. But when 
a' and 6* are commensurable, two or more fundamental modes 
may have the same periodic time, and may then coexist in any 
proportions, while the motion still retains its simple harmonic 
character. In such cases the specification of the period does 
not completely determine the type. The full consideration of 
the problem now presenting itself requires the aid of the theory 
of numbers; but it will be suflScient for the purposes of this 
work to consider a few of the simpler cases, which arise when 
the membrane is square. The reader will find fuller information 
in Riemann's lectures on partial differential equations. 



w 



312 VIBRATIONS OP MEMBRANES. [197. 

Ifa = 6, 

I'^j^^^ (=>■ 

The lowest tone is found by putting m and n equal to unity, 
which gives only one fundamental mode : — 

. irx , iry ^ 
w — svcL — sm — co8j)f (3). 

Next suppose that one of the numbers m, n is equal to 2, and 
the other to unity. In this way two distinct types of vibration 
are obtained, whose periods are the same. If the two vibrations 
be synchronous in phase, the whole motion is expressed by 

= -^C;sm sin -^ + 2) sin — sm — ^Vcospf...(4); 

so that, although every part vibrates synchronously with a 
harmonic motion, the type of vibration is to some extent arbitrary. 
Four particular cases may be especially noted. First, if 2) = 0, 

^ . 27ra? . ^ ^ 
w^ 1/ sm sin— ^ cos of (6), 

which indicates a vibration with one node along the line x = ^a. 
Similarly if (7 = 0, we have a node parallel to the other pair of 
edges. Next, however, suppose that C and D are finite and 
equal. Then w is proportional to 

. 27ra? . TTV . TTX . 27ry 

sin sm - - + sm sm — - , 

a a a a 

which may be put into the form 

^ . TTX . Try / 7ra? . iry\ 

2 sm — sm — ^ cos h cos — ^ . 

a a \ a aj 

This expression vanishes, when 

sin TTx/a = 0, or sin Try/a = 
or again, when 

cos TTx/a + cos Try /a = 0. 

The first two equations give the edges, which were originally 
assumed to be nodal ; while the third gives y + a? = a, representing 
one diagonal of the square. 

In the fourth case, when C = — 2), we obtain for the nodal 
lines, the edges of the square together with the diagonal y = x. 
The figures represent the four cases. 



197.] 



CASES OF EQUAL PEBIOD& 



313 



D = 


= 0. 







Fig. 82. 

(7 = 0. C-D = 0. a + 2) = o. 





[Frequency (referred to gravest) = 1*58.] 

For other relative values of C and D the interior nodal line 
is curved, but is always analytically expressed by 



Ccos— + 2)cos^ = 
a a 



(6), 



and may be easily constructed with the help of a table of logarith- 
mic cosines. 

The next case in order of pitch occurs when m = 2, n = 2. 

The values of m and n being equal, no alteration is caused by 

their interchange, while no other pair of values gives the same 

frequency of vibration. The only type to be considered is 

accordingly 

. 27rx . 27rv 



k; = sm sm — *^ cos pt, 

a a ^ 



Fig. 83. 



whose nodes, determined by the equation 

. TTX . wy irx Try ^ 
sm — sin — ^ cos — cos -^ = 0, 
a a a a 

are (in addition to the edges) the straight lines 

Fig. (33) 

x = \a y = ia. 

[Frequency = 200.] 

The next case which we shall consider is obtained by ascribing 
to m, n the values 3, 1, and 1, 3 successively. We have 




w 



{^ . ^irx .Try y. . irx , Sttv) 

= xC/sm sm — ^ + i)sm — sm — - \ cos pt 

{ a a a a ) ^ 



The nodes are given by 
8in^8in^|(7(4co8«^-l) + Z)(4co8«^-l)l = 0. 
or, if we reject the first two fsictors, which correspond to the edges. 

(7(4co8»^'^-l)+2)(4cos'^-l) = (7). 



814 VIBRATIONS OF MEMBBAKES. 

If £7 = 0, we have y = ^a, y = ^a. 
Ifi> = 0, a!=ia, x^ia. 

a a 

whence, y = iCi y = a — a, 

which represent the two diagonals. 

Lastly, if (7s= D, the equation of the node is 

cos' h coe' -^ = J, 



[197. 



1 + ci 



.(8). 



0=0. 




I)= 


). 




0+J)-0 




c-i)-o. 














\ / 

A 




o 





[Frequency = 2'24.] 

In case (4) when x^^a, y = ^a, orja; and similarly when 
y = ia, a;=ijr, or Jo. Thus one half of each of the lines joining 
the middle points of opposite edges ia intercepted by the curve. 

[The diameters of the nodal curve parallel to the sides of the 
square are thus equal to ^a. Those measured along the diagonals 
are sensibly smaller, equal to J\/2 . a, or '471 a.] 

It should be noticed that in whatever ratio to one another 
C and D may be taken, the nodal curve always passes through 
the four points of intersection of the nodal lines of the first two 
cases, G = 0, D^O. If the vibrations of these 
cases be compounded with corresponding phases, 
it is evident that in the shaded compartments of 
Fig. (35) the directions of displacement are the 
same, and that therefore no part of the nodal curve 
is to be found there ; whatever the ratio of ampli- 
tudes, the curve must be drawn through the un- 
shaded portions. When on the other hand the phases are opposed, 
the nodal curve will pass exclusively through the shaded portions. 



Fie. 86. 




197. J EFFECT OF SLIGHT IRREGULARITIES. 315 

When m = 3, n = 3, the nodes are the straight lines parallel 
to the edges shewn in Fig. (36). 

Fig. 36. 

The last case [Frequency = 2*55] which we 
shall consider is obtained by putting 



" 1 I "" 



— J — ^ — 

i i 



m = 3, n = 2, or m = 2, n=3. 

The nodal system is 

[Frequency = 300.] 

^ . Sirx , 2'iry r\ • 27ra? . 37ry . 

C/Sin sm — -+D8m — sm =0, 

a a a a 

or, if the factors corresponding to the edges be rejected, 

Cf4cos'^-l')cos^ + 2)cos'^f4cos«^-0 = (9). 

If C or i) vanish, we fall back on the nodal systems of the 
component vibrations, consisting of straight lines parallel to the 
edges. If C = 2), our equation may be written 

(cos — + cos ^ ) (4 cos — cos ^ - 1 ) = (10), 

\a a/\ a a I 

of which the first factor represents the diagonal y + ^^^a, and 
the second a hyperbolic curve. 

If C = — i), we obtain the same figure relatively to the other 
diagonals 

198. The pitch of the natural modes of a square membrane, 
which is nearly, but not quite uniform, may be investigated by 
the general method of § 90. 

We will suppose in the first place that m and n are equal. 
In this case, when the pitch of a uniform membrane is given, 
the mode of its vibration is completely determined. If we now 
conceive a variation of density to ensue, the natural type of 
vibration is in general modified, but the period may be calculated 
approximately without allowance for the change of type. 

We have 

T = \ ffipo + Bp) «^^,„' sin' '—■ sin' ^ cUcdy 

= i ^mm' jpo ^ + jjSp 8in« ^ 8in« ^ dxdyl , 

1 Lain£, Lefotu tur VtUutieiti, p. 129. 



316 VIBRATIONS OF MEMBRANES. [198. 

of which the second term is the increment of T due to ip. Hence 
if w oc cospf, and P denote the value of p previously to variation, 
we have 

i>»»*:P„«' = l-^J"r^8in«^J?sin»^<fo:dy (1). 

where Pmm''^ — ^^ , and d'^^T^-r-p^. 

For example, if there be a small load M attached to the middle of 
the square, 

/>mm^:-Pmm*=l-"j- siu* m ^ (2), 

ctrpo z 

in which sin* ^mir vanishes, if m be even, and is equal to unity, if 
m be odd. In the former case the centre is on the nodal line of 
the unloaded membrane, and thus the addition of the load produces 
no result. 

When, however, m and n are unequal, the problem, though re- 
maining subject to the same general principles, presents a pecu- 
liarity different from anything we have hitherto met with. The 
natural type for the unloaded membrane corresponding to a speci- 
fied period is now to some extent arbitrary ; but the introduction 
of the load will in general remove the indeterminate element. In 
attempting to calculate the period on the assumption of the undis- 
turbed type, the question will arise how the selection of the undis- 
turbed type is to be made, seeing that there are an indefinite 
number, which in the uniform condition of the membrane give 
identical periods. The answer is that those t}'pes must be chosen 
which differ infinitely little from the actual types assumed under 
the operation of the load, and such a type will be known by the 
criterion of its making the period calculated from it a maximum 
or minimum. 

As a simple example, let us suppose that a small load M is 
attached to the membrane at a point lying on the line a = Ja, and 
that we wish to know what periods are to be substituted for the 
two equal periods of the unloaded membrane, found by making 

m=2, w=l, or m = l, n=2. 

It is clear that the normal types to be chosen, are those whose 
nodes are represented in the first two cases of Fig. (32). In the 
first case the increase in the period due to the load is zero, which 
is the least that it can be; and in the second case the increase 



198.] SOLUTIONS APPLICABLE TO A TRIANGLE. 317 

is the greatest possible. If P be the ordinate of M, the kiuetic 
energy is altered in the ratio 

2 4*24^2^^'' a ' 

and thus p^^ : P„«= 1 -*f sin"-'"-'^ (3) 

^ a^p a ^ ^ 

while p^^ = P„» = P„». 

The ratio characteristic of the interval between the two natural 
tones of the loaded membrane is thus approximately 

1+ - sin» — '^ (4). 

a^p a ' 

If ^ = i a, neither period is affected by the load. 

As another example, the case where the values of m and n 
are 3 and 1, considered in § 197, may be referred to. With a load 
in the middle, the two normal types to be selected are those 
corresponding to the last two cases of Fig. (34), in the former 
of which the load has no effect on the period. 

The problem of determining the vibration of a square mem- 
brane which carries a relatively heavy load is more difficult, and 
we shall not attempt its solution. But it may be worth while to 
recall to memory the fact that the actual period is greater than 
any that can be calculated from a hypothetical type, which differs 
from the actual one. 

199. The preceding theory of square membranes includes a 
good deal more than was at first intended. Whenever in a vibrat- 
ing system certain parts remain at rest, they may be supposed to 
be absolutely fixed, and we thus obtain solutions of other questions 
than those originally proposed. For example, in the present case,, 
wherever a diagonal of the square is nodal, we obtain a solution 
applicable to a membrane whose fixed boundary is an isoscelea 
right-angled triangle. Moreover, any mode of vibration possible to 
the triangle corresponds to some natural mode of the square, as 
may be seen by supposing two triangles put together, the vibra- 
tions being equal and opposite at points which are the images of 
each other in the common h3rpothenuse. Under these circum- 
stances it is evident that the hypothenuse would remain at rest 
without constraint, and therefore the vibration in question is in- 
cluded among those of which a complete square is capable. 



(2), 



318 VIBRATIONS OF MEMBRANES. [199. 

The frequency of the gravest tone of the triangle is found by 
putting m = 1, n = 2 in the formula 

^ = ^V(m' + n') (1). 

and is therefore equal to c^Jo/ia. 

The next tone occurs, when m = 3, w = 1. In this case 

27r" 2a 

as might also be seen by noticing that the triangle divides itself 

into two, Fig. (37), whose sides are less 

than those of the whole triangle in the ^' ^^* 

ratio ^2 : 1. yi\ 

For the theory of the vibrations of 
a membrane whose boundary is in the 
form of an equilateral triangle, the / j \ 

reader is referred to Lamp's Lefons 

sur Vilasticiti. It is proved that the frequency of the gravest 
tone is c -h A, where h is the height of the triangle, which is the 
same as the frequency of the gravest tone of a square whose 
diagonal is h 

200. When the fixed boundary of the membrane is circular, 
the first step towards a solution of the problem is the expression 
of the general diflferential equation in polar co-ordinates. This 
may be effected analytically ; but it is simpler to form the polar 
equation de novo by considering the forces which act on the polar 
element of area rdOdr. As in § 194 the force of restitution acting 
on a small area of the membrane is 

and thus, if TJp = c* as before, the equation of motion is 

dp 

The subsidiary condition to be satisfied at the boundary is that 
w = 0, when r = a. 

In order to investigate the normal component vibrations we 
have now to assume that te; is a harmonic function of the time. 



w _ (d^w 1 ^ , J^ c^w) 
f^idi^'^rd^'^^dd^] ^^^- 



200.] POLAR CO-ORDINATES. 319 

Thus, i{ w QC cos (pt — e), and for the sake of brevity we write 
p/c = k, the differential equation appears in the fonn 

d^w Idw 1 d^w . ,, ^ .rt\ 

d^ + rd; + r»d^ + *'«'=^^ <2). 

in which k is the reciprocal of a linear quantity. 

Now whatever may be the nature of k; as a function of r and 0, 
it can be expanded in Fourier's series 

w=^Wo + Wi cos (0 + tti) + Wj cos 2 (^ + a,) + (3), 

in which Wq, Wi, &c. are functions of r, but not of 0. The result 
of substituting from (3) in (2) may be written 

the summation extending to all integral values of n. If we 
multiply this equation by cos n (d + On), and integrate with respect 
to between the limits and 27r, we see that each term must 
vanish separately, and we thus obtain to determine t(;n as a 
function of r 

d^Wn . 1 dwn 



+ -,^^ + (^-^)^n = (4), 



dr^ 

in which it is a matter of indifference whether the factor 
cos n (0 + On) be supposed to be included in Wn or not. 

The solution of (4) involves two distinct functions of r, 
each multiplied by an arbitrary constant. But one of these 
functions becomes infinite when r vanishes, and the correspondiDg 
particular solution must be excluded as not satisfying the pre- 
scribed conditions at the origin of co-ordinates. This point may 
be illustrated by a reference to the simpler equation derived from 
(4) by making k and n vanish, when the solution in question 
reduces to w^logr, which, however, does not at the origin 
satisfy V'w = 0, as may be seen from the value of f(dw/dn) ds, inte- 
grated round a small circle with the origin for centre. In like 
manner the complete integral of (4) is too general for our 
present purpose, since it covers the case in which the centre of 
the membrane is subjected to an external force. 

The other function of r, which satisfies (4), is the Bessel's 
function of the rfi^ order, denoted by Jn (kr), and may be expressed 
in several waya The ascending series (obtained immediately 
from the differential equation) is 



y 



1 



•/ .. ' 



w 



r 



320 VIBRATIONS OP MEMBRANES. [200. 



«^n(^)=2»r(ii+i)r"2: 



z^ 



2n + 2 2. 4. 2n+ 2.271 + 4 



+ 



} (5), 



2.4.6.2n + 2.2n-l-4.2n + 6 
from which the following relations between functions of consecu- 
tive orders may readily be deduced : 

/o'(^) = -/i(^) (6), 

2//(^) = /n-iW-/n+i(^) (7), 

^j„(z) = j;^,(^) + j„+.(^) (8). 

z 

When n is an integer, J^ {z) may be expressed by the definite 
integral 

1 r» 

Jn{^)=^ - I cos (i: sin O) — 710)) do) (9), 

which is Bessel's original form. From this expression it is evident 
that Jn and its differential coefficients with respect to z are always 
less than unity. 

The ascending series (6), though infinite, is convergent for all 
values of n and z\ but, when z is great, the convergence does not 
begin for a long time, and then the series becomes useless as a 
basis for numerical calculation. In such cases another series 
proceeding by descending powers of z may be substituted with 
advantage. This series is 

r/x /^Ii (1" - 4r't") (3« - 47i«) ^ \ ( IT ir\ 

-^-(^)=V^i^^ 1.2.(8.)' ^ r^(^-4"^2) 

/T { V - 4n« (1« - 47i») (3' " 4n») (5' - 47i') 1 

^V -rrzX \.Sz 1.2.3.(8.)» ^ | 

xsin^.-'^-7i|) (10); 

it terminates, if 2n be equal to an odd integer, but otherwise, it 
runs on to infinity, and becomes ultimately divergent. Nevertheless 
when z is great, the convergent part may be employed in calcula- 
tion ; for it can be proved that the sum of any number of terms 
differs from the true value of the function by less than the last 
term included. We shall have occasion later, in connection with 
another problem, to consider the derivation of this descending series. 
As Bessel's functions 6u:e of considerable importance in theo- 
retical acoustics, I have thought it advisable to give a table for 
the functions J^ and t/i, extracted from Lommel's^ work, and due 

1 Loxxunel, Studien iiber die BetteVschen Functionen, Leipzig, 1868. 



200.] 



BESSELS FUNCTIONS. 



321 



originally to Hansen, 
the relation 



The functions J^ and Ji are connected by 



z 
00 


^o(^) 


J^{z) : 


z 
45 


Jo{^) 


J,{z) 


Z Jo (2) 

1 


1 


10000 


00000 


•3205 


•2311 


9^0 • 


0903 


•2453 


01 


•9975 


•0499 ; 


4 6 


•2961 


•2566 


9^1 


1142 


•2324 


0-2 


•9900 


•0995 


4 7 


•2693 


•2791 


9^2 


•1367 


•2174 


0-3 


•9776 


•1483 


48 


•2404 


•2985 


93 


1577 


•2004 


0-4 


•9604 


•1960 


4-9 


•2097 


•3147 


9^4 


1768 


•1816 


0-5 


•9385 


•2423 


5 


•1776 


, 3276 


9-5 


1939 


•1613 


0-6 


•9120 


•2867 


5^1 


•1443 


•3371 


9-6 


•2090 


•1395 


07 


•8812 


•3290 


52 


•1103 


•3432 


9^7 


•2218 


•1166 


0-8 


•8463 


•3688 


5-3 


•0758 


•3460 


9^8 


•2323 


•0928 


0-9 


•8075 


•4060 


5-4 


•0412 


•3453 


9^9 ' 


•2403 


•0684 


1-0 


•7652 


•4401 


5-5 


•-•0068 


•3414 


10-0 


•2459 


•0435 


M 


•7196 


•4709 


5^6 


+ ^0270 


•3343 


10-1 


•2490 


+ •0184 


1-2 


•6711 


•4983 


5-7 


•0599 


•3241 


10^2 


•2496 


-■0066 


1-3 


•6201 


•5220 


58 


•0917 


•3110 


10^3 


•2477 


•0313 


1-4 


•5669 


•5419 : 


6-9 


•1220 


•2951 


10^4 


•2434 


•0555 


1-5 


•5118 


•5579 


6-0 


•1506 


•2767 


10^5 


•2366 


■0789 


1-6 


•4554 


•5699 


6-1 


•1773 


•2559 


10^6 • 


2276 


•1012 


1-7 


•3980 


•5778 


6-2 


•2017 


•2329 


10-7 


•2164 


•1224 


1-8 


•3400 


•5815 


6 3 


•2238 


•2081 


10^8 


2032 


•1422 


1-9 


•2818 


•5812 


6^4 


•2433 


•1816 


10^9 • 


1881 


•1604 


20 


•2239 


•5767 


65 


•2601 


•1538 ' 


110 • 


1712 


•1768 


2-1 


•1666 


•5683 


6^6 


•2740 


•1250 


111 • 


1528 


•1913 


2-2 


•1104 


•5560 


6-7 


•2851 


•0953 


11-2 • 


1330 


•2039 


2-3 


•0555 


•5399 


6^8 


•2931 


•0652 


ir3 • 


1121 


•2143 


2-4 


+ ^0025 


•5202 


69 


•2981 


•0349 


11-4 • 


0902 


•2225 


2-5 


- ^0484 


•4971 i 


7^0 


•3001 


- ^0047 


11^5 


0677 


•2284 


2-6 


•0968 


•4708 


7^1 


•2991 


+ ■0252 


116 • 


0446 


•2320 


2-7 


•1424 


•4416 


7-2 


•2951 


•0543 


117 -■ 


0213 


•2333 


2-8 


•1850 


•4097 


73 


•2882 


•0826 


11-8 +• 


0020 


•2323 


2-9 


. -2243 


•3754 


7^4 


•2786 


•1096 


11-9 • 


0250 


•2290 


3 


•2601 


•3391 


7 5 


•2663 


•1352 


12^0 • 


0477 


•2234 


31 


•2921 


•3009 


76 


•2516 


•1592 


12^1 


0697 


•2157 


3-2 


•3202 


•2613 


77 


•2346 


•1813 , 


122 • 


0908 


•2080 


3-3 


•3443 


•2207 1 


7^8 


•2154 


•2014 1 


123 • 


1108 


•1943 


3-4 


•3643 


•1792 


7-9 


•1944 


•2192 


12^4 


1296 


•1807 


3-5 


•3801 


•1374 


8-0 


•1717 


•2346 


12-5 


1469 


•1655 


3-6 


•3918 


•0955 


8-1 


•1475 


•2476 


12-6 • 


1626 


•1487 


3-7 


•3992 


•0538 


8^2 


•1222 


•2580 


12-7 • 


1766 


•1307 


3-8 


•4026 


+ ^0128 


83 


•0960 


•2657 


12^8 


1887 


•1114 


3-9 


•4018' 


-•0272 


8^4 


•0692 


•2708 


12^9 • 


1988 


•0912 


40 


•3972 


•0660 


8-5 


•0419 


•2731 


130 • 


2069 


•0703 


1 4.1 


•3887 


•1033 


8-6 


+ •0146 


•2728 


13^1 


2129 


•0489 


4-2 


•3766 


. •1386 


8^7 


-•0125 


•2697 


132 • 


2167 


•0271 


4-3 


•3610 


i ^1719 


8^8 


•0392 


•2641 ; 


13-3 • 


2183 


-•0052 


4-4 


•3423 


•2028 


8-9 


•0653 


•2559 


13^4 


2177 


+ •0166 



K. 



21 



^ 



322 VIBRATIONS OF MEMBRANES. [201. 

201. In accordance with the notation for BesseFs functions 

the expression for a normal component vibration may therefore be 

written 

w = PJn(kr) cosn(0 + a) cos(/)^ + €) (1); 

and the boundary condition requires that 

Jn{ka) = (2), 

an equation whose roots give the admissible values of k, and 
therefore of p. 

The complete expression for w is obtained by combining the 
particular solutions embodied in (1) with all admissible values of 
k and w, and is necessarily general enough to cover any initial 
circumstances that may be imagined. We conclude that any 
function of r and may be expanded within the limits of the 
circle r=a in the series 

w = 22 Jn (At) {<^ cos wd + -^ sin nd} (3). 

For every integral value of n there are a series of values of k, 
given by (2); and for each of these the constants <f> and yfr are 
arbitrary. 

The determination of the constants is effected in the usual 
way. Since the energy of the motion is equal to 



^pT Tw^rdedr (4), 

J J 



and when expressed by means of the normal co-oixlinates can only 
involve their squares, it follows that the product of any two of the 
terms in (3) vanishes, when integrated over the area of the circle. 
Thus, if we multiply (3) by Jn{kr)co8nd, and integrate, we 
find 



cos 710 rdrdd 



\ j wJnikr) 
JoJa 

= <^ f [ [Jn (kr)y COS' nd rdr dtf = <^ . tt T [J^ {h')Y rdr (o), 

by which ^ is determined. The corresponding formula for yjr is 
obtained by "writing sinn0 forcoswft A method of evaluating 
the integral on the right will be given presently. Since <f> and -^ 
each contain two terms, one varying as cosjp^ and the other as 
sinpt, it is now evident how the solution may be adapted so as to 
agree with arbitrary initial values of w and w. 



202.] CIRCULAR BOUNDARY. 323 

202. Let us now examine more particularly the character of 

the fundamental vibrations. If n = 0, w is a function of r only, 

that is to say, the motion is symmetrical with respect to the centre 

of the membmne. The nodes, if any, are the concentric circles, 

whose equation is 

Jo(kr)^0 (1). 

When 71 has an integral value different from zero, w is a func- 
tion of ^ as well as of r, and the equation of the nodal system 
takes the form 

Jn(kr) cos 71 (^- a) = (2). 

The nodal system is thus divisible into two parts, the first con- 
sisting of the concentric circles represented by 

Jn{fcr) = (3), 

and the second of the diameters 

^ = a + (2m + l)7r/27i (4), 

where 7/i is an integer. These diameters are n in number, and 
are ranged uniformly round the centre; in other respects their 
position is arbitrary. The radii of the circular nodes will be 
investigated further on. 

203. The important integral formula 

'jn(kr)Jn(k'r)rdr = (1), 



/, 





where k and k' are different roots of 

Jn{ka)^0 (2), 

may be verified analytically by means of the differential equations 
satisfied by Jn{kr), Jn{k'r); but it is both simpler and more 
instructive to begin with the more general problem, where the 
boundary of the membrane is not restricted to be circular. 

The variational equation of motion is 

SV+pjjwSwdxdy=-0 (3) 

where 

^'i'M^hm'"'^ '«• 



and therefore 



sir T ffidwdSw dwdBw] , , ... 



21—2 



//• 



324 VIBRATIONS OF MEMBRANES. [203. 

In these equations t£; refers to the actual motion, and Bwtoa, hypo- 
thetical displacement consistent with the conditions to which the 
system is subjected. Let us now suppose that the system is exe- 
cuting one of its normal component vibrations, so that w=^u, and 

u + p^u^O (6), 

while Bw is proportional to another normal function v. 

Since k^pjc, we get from (3) 

A?//«i;(i^dy=//g| + ||}d^dy (7). 

The integral on the right is symmetrical with respect to n and v, 
and thus 

(k'^-k')jjuvdxdy = (8), 

where A'* bears the same relation to v that A-* bears to u. 

Accordingly, if the normal vibrations represented by u and v 
have different periods, 

uvdwdy = (9). 

In obtaining this result, we have made no assumption as to the 
boundary conditions beyond what is implied in the absence of re- 
actions against acceleration, which, if they existed, would appear 
in the fundamental equation (3). 

If in (8) we suppose k' = k, the equation is satisfied identically, 
and we cannot infer the value of jlu^chdt/. In order to evaluate 

this integral we must follow a rather different course. 

If u and V be functions satisfying within a certain contour the 
equations V*u + k^u = 0, V^v -|- A'^y = 0, we have 

(k''''-k')jjuvdxdy^ jj(vV''u-uV''v)dxdy 

tf du dv\ , 
'j['dJi-''d-n}'^ (1<^>' 

by Green s theorem. Let us now suppose that v is derived from 
u by slightly varying k, so that 

V = 1/ + ^ SA, k'^^k + Bk'y 
substituting in (10), we find 

j*//„.d..,./(ii-.^)* ("); 



203.] VALUES OF INTEGRATED SQUARES. 325 

or, if u vanish on the boundary, 

2kffu^d.a,=f£'£ds (12). 

For the application to a circular area of radius r, we have 

u ^ C08 nd Jn{kr)) .-^. 

v = cos7idJn{k'r)] ^ ""^^ 

and thus from (10) on substitution of polar co-ordinates and integra- 
tion with respect to 0, 

(k'^-'k')rJn{kr)Jn{k'r)rdr . . 

Jo 

= rJ„(k'r)^-Jn(kr)-rJ„(kr)^J„ik'r) (14). 

Accordingly, if 

and k and k' be different. 



rJn(kr)Jn{k'r)rdr^O (15), 





an equation first proved by Fourier for the case when 

J^ (At) =: Jn {k'r) = 0. 
Again from (11) 

dJdJ . dV 

dJc 



2A* I t/n* (kr) rdr = r ^ -^ rJ^- 

Jq ^ ^ dkdr dr 



= jfcr> J'a - ki^'J 



dashes denoting differentiation with respect to kr. Now 



and thus 



2^y^^{kr)rdr^r-J,;-{kr)-\-r^{\ - J^l^Jn'Ocr) (16). 

This result is general ; but if, as in the application to membranes 
with fixed boundaries, /„ {kr) = 0, 

then 2 \'jn^{kr)rdr^r^J^^{kr) (17). 



326 VIBRATIONS OF MEMBRANES. [204. 

204. We may use the result just arrived at to simplify the 
expressions for T and V. From 

M; = 22 {<l>mnJ'n (JCmnr) COS uO + '>^mnJn {^Cmnr) siu 1x6] (1), 

we find 

r= i pira' 22 JnHKnO) [^mn^ + ^^n'] (2), 

F = i pira^. Xlp,nn'Jn' (^mn<0 [<l>mn' + >^,mr. (3) ; 

whence is derived the normal equation of motion 



2 4>«nn 
4>mn + Ihnn' 4>mn = ^QJ. (J,^;^a) ^^^' 

and a similar equation for '>^mn* The value of O^n is to be found 
from the consideration that ^mni4*mn denotes the work done by the 
impressed forces during a h3rpothetical displacement Z4>mn \ so that 
if Z be the impressed force, reckoned per unit of area, 



^»in= \\ZJn{^mn'^*) CQsnOrdrdO (5). 



These expressions and equations do not apply to the case n = 0, 
when (f> and yft are amalgamated. We then have 

T^if}7ra^Jo''(k^a)4>,no' I .^. 

r^ip7ra'Pmo'Jo'{kmoa)<l>mir' ^ ^' 

As an example, let us suppose that the initial velocities are zero, 
and the initial configuration that assumed under the influence of a 
constant pressure Z ; thus 

*^o -Z,2'rr \ Jo (kmor) rdr. 
Jo 

Now by the differential equation, 

rJ, (kr) = - (r Jo" (kr) + i Jo' (*r)}, 
and thus 



f 

Jo 



» a 



Jo{kr)rdr = ''^Jo'{ka) (8) 



SO that 4>,„^ = - -J- ZJo (k^n^a). 



'IM 



Substituting this in (7), we see that the initial value of ^mo is 

-2Z 

(<^mo)<-o = h~:;r'2-^-r.r—. (9). 



204.] SPECIAL PKOBLEMS. 327 



'inn 



For values of n other than zero, 4> and the initial value of <f> 
vanish. The state of the system at time t is. expressed by 

^ino = (^mo)f-o • COS Pmot '(10)» 

tO==l(f>,noJo{kimr) (11), 

the summation extending to all the admissible values of /r^. 

As an example o( forced vibrations, we may suppose that Z, still 
constant with respect to space, varies as a harmonic function of the 
time. This may be taken to represent roughly the circumstances 
of a small membrane set in vibration by a train of aerial waves. 
If Z= cos qt, we find, nearly as before, 

■"pa^^^^ k,no{q*-Pfno^)Jo(k„^a) 

The forced vibration is of course independent of 0. It will be seen 
that, while none of the symmetrical normal components are missing, 
their relative importance may vary greatly, especially if there be a 
near approach in value between q and one of the series of quanti- 
ties ptno- If the approach be very close, the effect of dissipative 
forces must be included. 

[Again, suppose that the force is applied locally at the centre. 

By (5) 

*„u, = -?i cos 2^ (13), 

if Zi cos qt denote the whole force at time t From (7) 

^ ^1 cos g^ n A\ 

and vj is then given by (11). The series is convergent, unless 
r = 0. 

But this problem would be more naturally attacked by including 
in the solutions of (4) § 200 the second Bessel's function § 341. 
In this method k = q/c ; and the ratio of constants by which the 
two functions of ?' are multiplied is determined by the boundary 
condition. When q coincides with one of the values of p, the 
second function disappears from the solution.] 

206. The pitches of the various simple tones and the radii of 
the nodal circles depend on the roots of the equation 

Jn (ka) = Jn (z) = 0. 



328 VIBRATIONS OF MEMBRANES. [205. 

If these (exclusive of zero) taken in order of magnitude be 

called Zn^\ Zn^^\ Zn^*^ Zn^'^ , then the admissible values of jo 

are to be found by multiplying the quantities ^n<*' by cja. The 
particular solution may then be written 



...(1). 



W = Jn (zn,^'^ -) {-4n<" COS nO + 5n"" sin 71^} cos |- Zn^n - €«<*' 

The lowest tone of the group 7i corresponds to ^h"' 5 and since in 
this case Jn {zn^^ r/a) does not vanish for any value of r less than a, 
there is no interior nodal circle. If we put « = 2, J^ will vanish, 
when 

« (2) !. 3- ^ a) 

(jv 

Z ^* 

that is, when r = a ^-- , 

which is the radius of the one interior nodal circle. Similarly 
if we take the root ^n**^, we obtain a vibration with «— 1 nodal 
circles (exclusive of the boundary) whose radii are 

All the roots of the equation J^, {ha) = are real. For, if 
possible, let A;a == X + 1;^ be a root ; then A;'a = X — ifi is also a root, 
and thus by (14) § 203, 



^i\fi I Jn (kr) Jn (Mr) rdr = 0. 



Now Jn(kr)y Jn(k'r) are conjugate complex quantities, whose 
product is necessarily positive ; so that the above equation requires 
that either X or ^ vanish. That X cannot vanish appears from 
the consideration that if ka were a pure imaginary, each term of 
the ascending series for Jn would be positive, and therefore the 
sum of the series incapable of vanishing. We conclude that 
^ = 0, or that k is reaP. The same result might be arrived at 
from the consideration that only circular functions of the time 
can enter into the analytical expression for a normal component 
vibration. 

The equation Jn (z) = has no equal roots (except zero). From 
equations (7) and (8) § 200 we get 

Jn ^'zJni" Jn-¥\ > 
Z 

^ Riemann, Partielle Differentialgleichungen, Braanschweig, 1869, p. 2C0. 



205.] ROOTS OF BESSEL^S FUNCTIONS. 329 

whence we see that if Jn, Jn vanished for the same value of z, Jn+i 
would also vanish for that value. But in virtue of (8) § 200 
this would require that all the functions Jn vanish for the value 
of z in question \ 

206. The actual values of Zn may be found by interpolation 
from Hansen's tables so far as these extend ; or formule may be 
calculated from the descending series by the method of successive 
approximation, expressing the roots directly. For the important 
case of the symmetrical vibrations (n = 0), the values of Zq may be 
found from the following, given by Stokes*: 

V"_ o. '050661 -053041 '2 62051 , 

V"^"''^'^"^'4^^n:'"(4*-i)»'^(4«-iy ^^^• 

For n = 1, the formula is 

^iW_ -151982 015399 -245270 

The latter series is convergent enough, even for the first root, 
corresponding to « = 1. The series (1) will suffice for values of 8 
greater than unity; but the first root must be calculated 
independently. The accompanying table (A) is taken from 
Stokes* paper, with a slight difference of notation. 

It will be seen either from the formulae, or the table, that the 
difference of successive roots of high order is approximately tt. 
This is true for all values of n, as is evident from the descending 
series (10) § 200. 

[The general formula, analogous to (1) and (2), for the roots of 
Jn {z) has been investigated by Prof. M^^Mahon. If m = 4n', and 

a = j7r(2u-l+4«) (3), 

, ,^, m-1 4(m-l)(7m-31) 
we have ..-=a- ^^ ^^^ 

32 (m - 1) (83?yt^ ~ 982m •+ 3779) 

io{Say ~ "^ ^*^- 

1 Bourget, " M^moire snr le mouvement vibratoire des membranes circulaires," 
Ann. de Vicole normale^ t. in., 1S66. In one passage M. Bourget implies that he 
has proved that no two Bessel's (unctions of integral order can have the same root, 
but I cannot find that he has done so. The theorem, however, is probably true ; 
in the case of functions, whose orders di£fer by 1 or 2, it may be easily proved from 
the formuliB of § 200. 

' Camh, Phil. Tram. Vol. ix. " On the numerical calculation of a class of defi- 
nite integrals and infinite series." [In accordance with the calculation of Prof. 
M<^Mahon the numerator of the last term in (2) has been altered from *245S3o 
to -245270.] 



330 



VIBRATIONS OF M£MBBAN£S. 



[206. 



This formula may be applied not only to integral values of n as in 
(1) and (2), but also when n is fractional. The cases of n = J, and 
n = f are considered in § 207.] 

M. Bourget has given in his memoir very elaborate tables of 
the frequencies of the different simple tones and of the radii of 
the nodal circles. Table B includes the values of z, which satisfy 
Jn(^), for n = 0, 1, ... 5, «= I, 2, ... 9. 

Table A. 



8 


-forJJz) = 0. 




V 


1 


•7655 


2 


1-7571 


3 


2-7546 


4 


3-7534 


5 


4-7527 


6 


5-7522 


7 


6-7519 


8 


7-7516 


9 


8-7514 


10 


9-7513 


11 


10-7512 


12 


11-7611 



Diff. 



-for.7,(«) = 0. 

IT 



•9916 
•9975 
•9988 
•9993 
•9995 
•9997 
•9997 
•9998 
•9999 
•9999 
-9999 



1-2197 

2-2330 

3-2383 

4-2411 

5-2428 

6-2439 

7-2448 

8-2454 

9-2459 

10-2463 

11-2466 

12-2469 



Diff. 



10133 
1-0053 
1-0028 
1-0017 
1-0011 
1-0009 
1-0006 
1-0005 
10004 
1-0003 
1-0003 



When n is considerable the calculation of the earlier roots 
becomes troublesome. For very high values of n, Zn^^^/n approxi- 
mates to a ratio of equality, as may be seen from the consideration 
that the pitch of the gravest tone of a very acute sector must tend 
to coincide with that of a long parallel strip, whose width is equal 
to the greatest width of the sector. 

Table B. 



8 


n = 


1 


2-404 


2 


5-520 


3 


8-654 


4 


11-792 


5 


14-931 


6 


18-071 


7 


21-212 


8 


24-353 


9 


27-494 




3-832 
7-016 
10-173 
13-323 
16-470 
19-616 
22-760 
25-903 
29-047 



5-135 

8-417 
11-620 
14-796 i 
17-960 ' 
21-117 
24-270 , 
27-421 ; 
30-571 I 



6-379 
9-760 
13-017 
16-224 
19-410 
22-583 
25-749 
28-909 
32-050 



n = 4 



n = 5 



7-586 
11-064 
14-373 
17-616 
20-827 
24-018 
27-200 
30-371 
33-512 



8-780 
12-339 
15-700 
18-982 
22-220 
25-431 
28-628 
31-813 
34-983 



206.] 



NODAL FIGURES, 



331 




i.OOO 





9,iS0 




3. a SB 




i.534 




2,653 






4.144 




2,9IP 




3,B00 
.878^639 




4,1^4 



The figures represent the more important normal modes of 
vibration, and the numbers affixed give the frequency referred to 



332 VIBRATIONS OF MEMBRANES. [206. 

the gravest as unity, together with the radii of the circular nodes 
expressed as fractions of the radius of the membrane. In the case 
of six nodal diameters the frequency stated is the result of a rough 
calculation by myself. 

The tones corresponding to the various fundamental modes of 
the circular membrane do not belong to a harmonic scale, but 
there are one or two approximately harmonic relations which may 
be worth notice. Thus 

^ X 1-594 = 2125 = 2-136 nearly, 
f X 1-594 = 2-657 = 2-653 nearly, 
2 X 1-594 = 3188 = 3*156 nearly; 

80 that the four gravest modes with nodal diameters only would 
give a consonant chord. 

The area of the membrane is divided into segments by the 
nodal system in such a manner that the sign of the vibration 
changes whenever a node is crossed. In those modes of vibration 
which have nodal diameters there is evidently no displacement of 
the centre of inertia of the membrane. In the case of symmetri- 
cal vibrations the displacement of the centre of inertia is propor- 
tional to 

|V. (kr) rdr = -£ U" (kr) + i J,' (kr) | rdr = - | J,' (ka), 

an expression which does not vanish for any of the admissible 
values of k, since Jq {z) and Jo (z) cannot vanish simultaneously. 
In all the symmetrical modes there is therefore a displacement of 
the centre of inertia of the membrane. 

207. Hitherto we have supposed the circular area of the 
membrane to be complete, and the circumference only to be 
fixed; but it is evident that our theory virtually includes the 
solution of other problems, for example — some cases of a mem- 
brane bounded by two concentric circles. The complete theory 
for a membrane in the form of a ring requires the second Bessel's 
function. 

The problem of the membrane in the form of a semi-circle 
may be regarded as already solved, since any mode of vibration 
of which the semi-circle is capable must be applicable to the 



207.] FIXED RADIUS. 333 

complete circle also. In order to see this, it is only necessary 
to attribute to any point in the complementary semi-circle the 
opposite motion to that which obtains at its optical image in 
the bounding diameter. This line will then require no constraint 
to keep it nodal. Similar considerations apply to any sector 
whose angle is an aliquot part of two right angles. 

When the opening of the sector is arbitrary, the problem 
may be solved in terms of Bessels functions of fractional order. 
If the fixed radii are tf = 0^ d= ^8, the particular solution is 

vtrO 
tv = PJy^fp{kr) sin ^ coe{pt-€) (1). 

where v is an integer. We see that if fi be an aliquot part of tt, 
V7r/I3 is integral, and the solution is included among those aheady 
used for the complete circle. 

An interesting case is when fi = 27r, which corresponds to the 
problem of a complete circle, of which the 
radius tf = is constrained to be nodal. ^'^* *®- 

We have 

w = PJ^^ (kr) sin ^p0 cos (pt — c). 

When V is even, this gives, as might be 
expected, modes of vibration possible without 
the constraint; but, when v is odd, new 
modes make their appearance. In fact, in 
the latter case the descending series for / 
tenninates, so that the solution is expressible in finite terms.. 

Thus, when j/ = 1, 

sm iw* 
w = P ^j^ sm^d cos (pt'-e) (2). 

The values of k are given by 

sin ka = 0, or ka = sir. 

Thus the circular nodes divide the fixed radius into equal 
parts, and the series of tones form a har- ^w, 39. 

monic scale. In the case of the gravest 
mode, the whole of the membrane is at any 
moment deflected on the same side of its 
equilibrium position. It is remarkable that 
the application of the constmnt to the 
radius ^ = makes the problem easier than 
before. 





334 VIBRATIONS OF MEMBRANES. [207. 

If we take j/ = 3, the solution is 

^ " ^ Mcr) Cl7~ "" ^^® *V ®^^ ^^ ^^^ ^^^ " ^^ ^^^' 

In this case the nodal radii are Fig. (39) 

and the possible tones are given by the equation 

tan ka = ka (4). 

To calculate the roots of tan a? = a? we may assume 

a: = (5 + i)'7r-y = X-y, 
where y is a positive quantity, which is small when x is large. 

Substituting this, we find cot y^^X-^y, 
whence 

^~XV Z Z' V 3 16 315 ••• 

This equation is to be solved by successive approximation. 
It will readilv be found that 

„ 2 ^ 13 ^ 146 ^ 

so that the roots of tan a? = a? are given by 

. = Z-Z— |z-.-.lfz--^«X-- (5). 

where X = (8 + i) tt. 

In the first quadrant there is no root after zero since tan x>x, 
and in the second quadrant there is none because the signs of 
X and tana; are opposite. The first root after zero is thus in 
the third quadra it, corresponding to «=1. Even in this case 
the series converges sufficiently to give the value of the root 
with considerable accuracy, while for higher values of 8 it is 
all that could be desired. The actual values of x/ir are 1*4303, 
2-4590, 3-4709, 4*4747, 5-4818, 6-4844, &c. 

^ 208. The effect on the periods of a slight inequality in the 
density of the circular membrane may be investigated by the 
general method § 90, of which several examples have already 
been given. It will be sufficient here to consider the case of a 



208.] EFFECT OF SMALL LOAD. 335 

small load M attached to the membrane at a point whose radius 
vector is r'. 

We will take first the symmetrical types (n = 0), which may 
still be supposed to apply notwithstanding the presence of M. The 
kinetic energy T is (6) § 204 altered from 

i pira' Jo'' (k^noo) <^,.o' to i pira' Jo'' (*moa) <l>mo' + i M4>,no' Jo' {kjnorl 
and therefore 

P^n, .imo-1 pjra^ J^'^k„,oa) ^^^' 

where P,,^' denotes the value o{ p,„a*, when there is no load. 

The unsymmetrical normal types are not fully determinate for 
the unloaded membrane ; but for the present purpose they must 
be taken so as to make the resulting periods a maximum or 
minimum, that is to say, so that the effect of the load is the 
greatest and least possible. Now, since a load can never raise 
the pitch, it is clear that the influence of the load is the least 
possible, viz. zero, when the type is such that a nodal diameter (it 
is indifferent which) passes through the point at which the load is 
attached. The unloaded membrane must be supposed to have two 
coincident periods, of which one is unaltered by the addition of the 
load. The other type is to be chosen, so that the alteration of 
period is as great as possible, which will evidently be the case 
when the radius vector / bisects the angle between two adjacent 
nodal diameters. Thus, if r coiTespond to ^ = 0, we are to take 

W = 4>^nn Jn {hnnT) COS uB ] 

so that (2) § 204 

r = i pTra^ <^,nn' Jn^ {hnnd) + i il/ <^,nu' Jn' (KnO. 

The altered pmn is therefore given by 

.. a . p 2 - 1 - ^^/- •^~' (^wnO /ox 

Of course, if r be such that the load lies on one of the nodal 
circles, neither period is affected. 

For example, let M be at the centre of the membrane. J„ (0) 
vanishes, except when 71 = ; and Jo (0) = 1. It is only the 
symmetrical vibrations whose pitch is influenced by a central load, 
and for them by (1) 

M 

Hmo ' ■'^ mo " ^ r '•»/;. «\ ^9 V*'/' 



336 VIBRATIONS OF MEMBRANES. [208. 

By (6) §200 J^{z) = ^J,{z\ 

so that the application of the formula requires only a knowledge of 
the values of Ji {z), when Jo {z) vanishes, § 200. For the gravest 
mode the value of Jo {kmoCt) is '51903*. When k^oO, is consider- 
able, 

Ji' (kmoo) = 2-5- Trk,„oa 

approximately; so that for the higher components the influence of 
the load in altering the pitch increases. 

The influence of a small irregularity in disturbing the nodal 
system may be calculated from the formulae of § 90. The most 
obvious effect is the breaking up of nodal diameters into curves 
of hyperbolic form due to the introduction of subsidiary sym- 
metrical vibrations. In many cases the disturbance is fiavoured 
by close agreement between some of the natural periods. 

209. We will next investigate how the natural vibrations of 
a uniform membrane are affected by a slight departure from the 
exact circular form. 

Whatever may be the nature of the boundary, w satisfies the 
equation 

d.- + r dr-^? d^+*^^ = (1), 

where fc is a constant to be determined. By Fourier's theorem w 
may be expanded in the series 

W==Wq'\- Wi cos (0 + tti) -I- Wa cos 2 (tf + Oa) -H 

+WnCosn(^+an) + 

where Wo, t^i, &c. are functions of r only. Substituting in (1), we 
see that Wn must satisfy 

dr* ^r dr ^V r^)'^'' ^' 

of which the solution is 

Wn 3C Jn (kr) ; 

for, as in § 200, the other function of r cannot appear. 

The general expression for w may thus be written 

w = AoJo {kr) + Ji (kr) (Ai cos d + Bi sin 0) 

+ ... + Jn(kr)(AnCO8ii0 + Bnsinn0) + (2). 

For all points on the boundary w is to vanish. 

1 The RQCceeding values are approximately '341, '271, '232, *206, '187, <Src. 



209.] NEAKLY CIRCULAR BOUNDARY. 337 

In the case of a nearly circular membrane the radius vector is 
nearly constant. We may take r==a + Br, Br being a small 
function of d. Hence the boundary condition is 

= Ao[Jo{ka) + kBrJo'(ka)] + 

+ [Jn (ka) + kBr Jn (ka)] [An cos nO + Bn sin nd] 

+ (3), 

which is to hold good for all values of 0. 

Let us consider first those modes of vibration which are nearly 
symmetrical, for which therefore approximately 

w — AqJq ikr). 

All the remaining coefficients are small relatively to ilo, since 
the type of vibration can only diflTer a little from what it would 
be, were the boundary an exact circle. Hence if the squares of 
the small quantities be omitted, (3) becomes 

A^[J^{ka)-¥kBrJ^\ka)]'¥Ji{ka)[AiQo&e-\-B^Axie] 

+ ... ^-Jnika) [ilnCosn^ + £nsinnd]+ ... =0 (4). 

If we integrate this equation with respect to 6 between the 
limits and 27r, we obtain 

27r Jo {ka) + J^ {ka) ( 'kBrdd = 0, 

Jo 



or 



Jo \ka + kj Br |^} = (5), 



which shews that the pitch of the vibration ia approximately the 
same as if the radius vector had uniformly its niean value. 

This result allows us to form a rough estimate of the pitch of 
any membrane whose boundary is not extravagantly elongated. 
If a- denote the area, so that pa is the mass of the whole mem- 
brane, the frequency of the gravest tone is approximately 



(27r)-^ X 2-404 x y ^^ (6)*. 



<rp 
In order to investigate the altered type of vibration, we may 

^ [A numerical error is here corrected.] 
R. 22 



338 VIBRATIONS OF MEMBRANES. [209. 

multiply (4) by cosn^, or sinnOy and then integrate as before. 
Thus 

Ao Jo{ka) I kBr cos nO d6 + irAn Jn (ka) = 

Jo 



y 



.(7), 



/•2Jr 

Ao Jo(ka) kBr sin 7i.d dO + -rrBn Jn (ka) = 
Jo 

which determine the ratios -4^ : Aq and Bn - Aq. 

If Sr=Sro + Sri-f ... + 8r„H- ... 

be Fourier's expansion, the final expression for w may be written, 
w :Ao = Jo (kr) 

^ k J,' (ka) )^-l^^ (8). 

When the vibration is not approximately symmetrical, the 
question becomes more complicated. The normal modes for the 
truly circular membrane are to some extent indeterminate, but the 
irregularity in the boundary will, in general, remove the indeter- 
minateness. The position of the nodal diameters must be taken, 
so that the resulting periods may have maximum or minimum 
values. Let us, however, suppose that the approximate tj'pe is 

w = A^J^{kr) cosi/d (9), 

and afterwards investigate how the initial line must be taken in 
order that this form may hold good. 

All the remaining coefficients being treated as small in com- 
parison with A,, we get from (4) 

Ao Jo (ka) + ... + Ay[Jy (ka) + khrJJ {ka)] cos vd 

-\-B^J^(ka)8inv0 -\- 

-hJnika) [AnCosn0-{'BnCosnd]'\- ,.. =0 (10). 

Multiplying by cos vd and integrating, 

J^ (ka) + k J J (ka) \ Br cos^ vBdO^O, 

Jo 



'o 
or 



pa + A? [ ' Sr cos^ pd —1 = 0, 



which shews that the effective radius of the membrane is 

r*' dO 
a+ Br cos^vO- (11). 



209.] NEARLY CIRCULAR BOUNDARY. 339 

The ratios of An and Bn to A^ may be found as before by in- 
tegrating equation (10) after multiplication by cos nO, sin nO. 

But the point of greatest interest is the pitch. The initial line 
is to be so taken as to make the expression (11) a maximum or 
minimum. If we refer to a line fixed in space by putting — a 
instead of 0, we have to consider the dependence on a of the 
quantity 

•2ir 

Sr cos^ viO-a) d0, 



J a 





which may also be written 

COS* I/a I Sr cos'^ i/^cW + 2 cos va sin va j Sr cos v0 sin v0d0 

Jo Jo 

-^-sin^par Srsm^v0d0 (12), 

Jo 

and is of the form 

A cos* pa -h 2-B cos pa sin i/a + C sin* i/a, 

A, By C being independent of a. There are accordingly two 
admissible positions for the nodal diameters, one of which makes 
the period a maximum, and the other a minimum. The diameters 
of one set bisect the angles between the diameters of the other 
set. 

There are, however, cases where the normal modes remain inde- 
terminate, which happens when the expression (12) is independent 
of a. This is the case when Sr is constant, or when Sr is propor- 
tional to cos p0. For example, if Sr were proportional to cos 20, 
or in other words the boundary were slightly elliptical, the nodal 
system corresponding to n = 2 (that consisting of a pair of per- 
pendicular diameters) would be arbitrary in position, at least to 
this order of approximation. But the single diameter, correspond- 
ing to 71=1, must coincide with one of the principal axes of 
the ellipse, and the periods will be different for the two axes. 

210. We have seen that the gravest tone of a membrane, 
whose boundary is approximately circular, is nearly the same as 
that of a mechanically similar membrane in the form of a circle of 
the same mean radius or area. If the area of a membrane be 
given, there must evidently be some form of boundary for which 
the pitch (of the principal tone) is the gravest possible, and this 

22—2 



340 VIBRATIONS OF MEMBRANES. [210. 

form can be no other than the circle. In the case of approximate 
circularity an analytical demonstration may be given, of which the 
following is an outline. 

The general value of w being 

w^AqJq (kr) + . . . + «/n {kr) {An cos nd + -B sin nff) + (1), 

in which for the present purpose the coefficients A^^Bi,. . . are small 
relatively to ilo, we find from the condition that w vanishes 
whenr = a+Sr, 

A^J^{ka)'\'kA^J,\ha)hrJr\l(^A^J^\ka).{irf^ 

+ 2 [{Jn (ifca) + ^n (*a) Sr + . ..}{il„ cos nfl + 5n sin nfl}] = 0... (2). 
Hence, if 

Sr = ai cos fl + A sill ^ +•••+ On cos nfl + /8n sin n^ + (3), 

we obtain on integration with respect to from to 27r, 



nas« 



24, J, + ^1^A, Jo" 2^^^ (o„» + ^„») 



n=« 



+ kl , [ia„An + finBn)Jn']=0 (4), 



n»l 



from which we see, as before, that if the squares of the small 
quantities be neglected, «7o (M^) — 0, or that to this order of ap- 
proximation the mean radius is also the effective radius. In 
order to obtain a closer approximation we first determine An : Aq 
and Bn : Aq by multipl3ring (2) by cos nfl, sin n0, and then in- 
tegrating between the limits and 27r. Thus 

Substituting these values in (4), we get 



J. (ka) = i A» ill' han" + /3„») l*^*^?' - i J,' j 
Since Jo satisfies the fundamental equation 



(6). 



J." + ^Jo' + Jo^O (7), 

and in the present case Jo = approximately, we may replace 
Jq' by — r- «/o'. Equation (6) then becomes 



l»=QO 



J.(te)=JA.J,'2^_^^(«„. + ^„»){^' + 2Ll 



(8). 



210.] FORM OF MAXIMUM PERIOD. 341 

Let 118 now suppose that a + da is the equivalent radius of the 
membrane, so that 

«/o [k (a + da)] = Jo (Jca) + J^ (ka) kda = 0. 

Then by (8) we find 

Again, if a + da' be the radius of the truly circular membrane 
of equal area, 



'^'=^cr<«»'+^»'> <i^>; 



so that 



The question is now as to the sign of the right-hand member. 
If n = 1, and z be written for fca, 

vanishes approximately by (7), since in general Ji = — Jq\ and 
in the present case Jq (z) = nearly. Thus da' — da = 0, as should 
evidently be the case, since the term in question represents merely 
a displacement of the circle without an alteration in the form of 
the boundary. When n = 2, (8) § 200, 

«/, — -t/i — «/o, 

Z 



from which and (7) we find that, when Jq = 0, 

Ja 2z 

whence 



(12), 



<ia'-da-^(a,'+/9,»)(^-l) (13), 



which is positive, since z = 2*404. 
We have still to prove that 

Jn (Z) 

is positive for integral values of n greater than 2, when z = 2*404. 



342 



VIBRATIONS OF MEMBRANES. 



[210. 



For this purpose we may avail ourselves of a theorem given in 
Biemann s Partielle Differentialgleichungen, to the effect that 
neither Jn nor Jn has a root (other than zero) less than w. The 
differential equation for «/« may be put into the form 

while initially Jn and Jn (as well as dJn/d logz) are positive. Ac- 
cordingly (Un/d log z begins by increasing and does not cease to do 
so before z = n, from which it is clear that within the range z = 
to z = n, neither Jn nor Jn can vanish. And since Jn and Jn are 
both positive until <e = n, it follows that, when n is an integer greater 
than 2*404, da' — da is positive. We conclude that, unless Oa, /Sj, 
as, ... all vanish, da' is greater than da, which shews that in the 
case of any membrane of approximately circular outline, the circle 
of equal area exceeds the circle of equal pitch. 

We have seen that a good estimate of the pitch of an approxi- 
mately circular membrane may be obtained from its area alone, 
but by means of equation (9) a still closer approximation may be 
effected. We will apply this method to the case of an ellipse, 
whose semi-axis major is jB and eccentricity e. 



The polar equation of the boundary is 
r = jB {l-.ic»-.^e* + +ie»cos2fl-h } 



(14); 



so that in the notation of this section 

o = ii(l-ie»-^e*), o, = ie»i2. 
Accordingly by (9) 



da = — ^ . kR . • 
or by (12), since kR = z = 2*404, 



[JA^)^2zr 



da — -- AT-^-R- 
o4 

Thus the radius of the circle of equal pitch is 

•779 e* 



a-hda = i?|l-.Te»-^^ 
( 4 64 



(15), 



in which the term containing e* should be correct. 



210.] ELLIPTICAL BOUNDARY. 343 

The result may also be expressed in terms of e and the area a. 
We have 



and thus 






from which we see how small is the influence of a moderate eccen- 
tricity, when the area is given. 

211. When the fixed boundary of a membrane is neither 
straight nor circular, the problem of determining its vibrations 
presents diflSculties which in general could not be overcome 
without the introduction of functions not hitherto discussed or 
tabulated. A partial exception must be made in favour of an 
elliptic boundary ; but for the purposes of this treatise the im- 
portance of the problem is scarcely sufficient to warrant the 
introduction of complicated analysis. The reader is therefore 
referred to the original investigation of M. Mathieu^ 

[The method depends upon the use of conjugate functions. If 

x + iy = eco8{^'¥ir)) (1), 

then the curves rj = const, are confocal ellipses, and f = const, are 
confocal hyperbolas. In terms of f, rj the fundamental equation 
(V'-hk^)u = becomes 

where k' = ke. 

The solution of (2) may be found in the form 

« = S(f).H(i,) (3). 

in which B is a function of f only, and H a function of rj only, 
provided 

^-(A:'»cos«f-a)B = (4), 

^'^ + (ik'»cosh»i7-a)H = (5), 

a being an arbitrary constant*. 

* Lioaville, xin., 1868 ; Court de phytique mathimatique, 1873, p. 122. 
- Pockels, Uber die partielle Differentialgleiehung Au+J;^u=0, p. 114. 



344 VIBRATIONS OF MEMBRANES. [211. 

Michell^ has shewn that the elliptic transformation (1) is the 
only one which yields an equation capable of satisfaction in the 
form (3).] 

Soluble cases may be invented by means of the general 
solution 

w — AqJq (At) + . . . + (il n cos n^ + ^n sin n fl) «7n (At) + 

For example we might take 

w=^Jq (kr) — X «/i (At) cos fl, 

and attaching different values to X, trace the various forms of 
boundary to which the solution will then apply. 

Useful information may sometimes be obtained from the 
theorem of § 88, which allows us to prove that any contraction of 
the fixed boundary of a vibrating membrane must cause an eleva- 
tion of pitch, because the new state of things may be conceived to 
differ from the old merely by the introduction of an additional 
constraint. Springs, without inertia, are supposed to urge the 
line of the proposed boundary towards its equilibrium position, 
and gradually to become stiffen At each step the vibrations 
become more rapid, until they approach a limit, corresponding to 
infinite stiffness of the springs and absolute fixity of their points 
of application. It is not necessary that the part cut off should 
have the same density as the rest, or even any density at all. 

For instance, the pitch of a regular polygon is intermediate 
between those of the inscribed and circumscribed circles. Closer 
limits would however- be obtained by substituting for the circum- 
scribed circle that of equal area according to the result of § 210. 
In the case of the hexagon, the ratio of the radius of the circle of 
equal area to that of the circle inscribed is 1*050, so that the mean 
of the two limits cannot differ from the truth by so much as 2^ per 
cent. In the same way we might conclude that the sector of a 
circle of 60^ is a graver form than the equilateral triangle obtained 
by substituting the chord for the arc of the circle. 

The following table giving the relative frequency in certain 
calculable cases for the gravest tone of membranes under similar 
mechanical conditions and of equal area (a), shews the effect of a 
greater or less departure from the circular form. 

^ Me$$enger of Mathematicit vol. xiz. p. 86, 1890. 



211.] MEMBRANES OF EQUAL AREA. 345 

Circle 2404 . Vtt = 4-261. 

Square V2 . -w = 4*443. 

Quadrant of a circle -^— . V7r = 4'551. 



v?= 



Sector of a circle 60^ 6-379 a/ ^ =4*616. 

Rectangle 3x2 A/-g- . tt = 4624. 

Equilateral triangle 27r.Vtan30^= 4*774. 

Semicircle 3*832 a/| = 4*803. 

Rectangle 2x1 ) ^^/I^^.qq^ 

Right-angled isosceles triangle J v 2 

lo 



Rectangle 3 x 1 7rW-^ = 5*736. 

For instance, if a square and a circle have the same area, the 
former is the more acute in the ratio 4*443 : 4*261, or 1*043 : 1. 

For the circle the absolute frequency is 

(27r)-» X 2*404 c a/- , where c = V2\ ■^ Vp- 

In the case of similar forms the frequency is inversely as the 
linear dimension. 

[From the principle that an extension of boundary is always 
accompanied by a fall of pitch, we may infer that the gravest 
mode of a membrane of any shape, and of any variable density, is 
devoid of internal nodal lines.] 

212. The theory of the free vibrations of a membrane was 
first successfully considered by Poisson^ His theory in the 
case of the rectangle left little to be desired, but his treatment 
of the circular membrane was restricted to the symmetrical 
vibrations. Kirchhoif's solution of the similar, but much more 
difficult, problem of the circular plate was published in 1850, 
and Clebsch's Theory of Elasticity (1862) gives the general theory 
of the circular membrane including the eflFects of stiffness and 

^ M£m. de VAcadimie, t. Tin. 1829. 



346 VIBRATIONS OF MEMBRANES. [212. 

of rotatory inertia^ It will therefore be seen that there was not 
much left to be done in 1866 ; nevertheless the memoir of Bourget 
already referred to contains a useful discussion of the problem 
accompanied by very complete numerical results, the whole of 
which however were not new. 

213. In his experimental investigations M. Bourget made use 
of various materials, of which paper proved to be as good as any. 
The paper is immersed in water, and after removal of the superfluous 
moisture by blotting-paper is placed upon a frame of wood whose 
edges have been previously coated with glue. The contraction of the 
paper in dr3dng produces the necessary tension, but many failures 
may be met with before a satisfactory result is obtained. Even 
a well stretched membrane requires considerable precautions in 
use, being liable to great variations in pitch in consequence of the 
varying moisture of the atmosphere. The vibrations are excited 
by organ-pipes, of which it is necessary to have a series proceeding 
by small intervals of pitch, and they are made evident to the eye 
by means of a little sand scattered on the membrane. If the 
vibration be suflSciently vigorous, the sand accumulates on the 
nodal lines, whose form is thus defined with more or less precision. 
Any inequality in the tension shews itself by the circles becoming 
elliptic. 

The principal results of experiment are the following : — 

A circular membrane cannot vibrate in unison with every sound. 
It can only place itself in unison with sounds more acute than 
that heard when the membrane is gently tapped. 

As theory indicates, these possible sounds are separated by less 
and less intervals, the higher they become. 

The nodal lines are only formed distinctly in response to 
certain definite sounds. A little above or below confusion ensues, 
and when the pitch of the pipe is decidedly altered, the membrane 
remains unmoved. There is not, as Savart supposed, a continuous 
transition from one system of nodal lines to another. 

The nodal lines are circles or diameters or combinations of 
circles and diameters, as theory requires. However, when the 

^ [The reader who vrishes to pursne the subject from a mathematical point of 
view is referred to an excellent discussion by Pockels (Leipzig, 1S91) of the 
differential equation ^hi + k^u aO.] 



213.] OBSERVATIONS OF M. BOURGET. 347 

number of diameters exceeds two, the sand tends to heap itself 
confusedly towards the middle of the membrane, and the nodes 
are not well defined. 

The same general laws were verified by MM. Bernard and 
Bourget in the case of square membranes*; and these author 
consider that the results of theory are decisively established in 
opposition to the views of Savart, who held that a membrane 
was capable of responding to any sound, no matter what its pitch 
might be. But I must here remark that the distinction between 
forced and free vibrations does not seem to have been sufficiently 
borne in mind. When a membrane is set in motion by aeiial 
waves having their origin in an organ-pipe, the vibration is 
properly speaking /arced. Theory asserts, not that the membrane 
is only capable of vibrating with certain defined frequencies, but 
that it is only capable of so vibrating freely. When however the 
period of the force is not approximately equal to one of the 
natural periods, the resulting vibration may be insensible. 

In Savart's experiments the sound of the pipe was two or three 
octaves higher than the gravest tone of the membrane, and was 
accordingly never far from unison with one of the series of over- 
tones. MM. Bourget and Bernard made the experiment under 
more favourable conditions. When they sounded a pipe somewhat 
lower in pitch than the gravest tone of the membrane, the sand 
remained at rest, but was thrown into vehement vibration as unison 
was approached. So soon as the pipe was decidedly higher than the 
membrane, the sand returned again to rest. A modification of the 
experiment was made by first tuning a pipe about a third higher 
than the membrane when in its natural condition. The membrane 
was then heated until its tension had increased sufficiently to 
bring the pitch above that of the pipe. During the process of 
cooling the pitch gradually fell, and the point of coincidence 
manifested itself by the violent motion of the sand, which at the 
beginning and end of the experiment was sensibly at rest. 

M. Bourget found a good agreement between theory and obser- 
vation with respect to the radii of the circular nodes, though the 
test was not very precise, in consequence of the sensible width of 
the bands of sand; but the relative pitch of the various simple 
tones deviated considerably from the theoretical estimates. The 

^ Ann. de Chim. Lx. 449—479. 1860. 



348 VIBRATIONS OF MEMBRANES. [213. 

committee of the French Academy appointed to report on 
M. Bourget's memoir suggest as the explanation the want of 
perfect fixity of the boundary. It should also be remembered that 
the theory proceeds on the supposition of perfect flexibility — a 
condition of things not at all closely approached by an ordinary 
membrane stretched with a comparatively small force. But 
perhaps the most important disturbing cause is the resistance of 
the air, which acts with much greater force on a membrane than 
on a string or bar in consequence of the large surface exposed. 
The gravest mode of vibration, during which the displacement is 
at all points in the same direction, might be affected very 
differently from the higher modes, which would not require so 
great a transference of air from one side to the other. 

[In the case of kettle-drums the matter is further complicated 
by the action of the shell, which limits the motion of the air upon 
one side of the membrane. From the fact that kettle-drums are 
struck, not in the centre, but at a point about midway between 
the centre and edge, we may infer that the vibrations which it is 
desired to excite are not of the symmetrical class. The sound is 
indeed but little affected when the central point is touched with 
the finger. Under these cii-cumstances the principal vibration (1) is 
that with one nodal diameter and no nodal circle, and to this 
corresponds the greater part of the sound obtained in the normal 
use of the instrument. Other tones, however, are audible, which 
correspond with vibrations characterized (2) by two nodal diameters 
and no nodal circle, (3) by three nodal diameters and no nodal 
circles, (4) by one nodal diameter and one nodal circle. By 
observation with resonators upon a large kettle-drum of 25 inches 
diameter the pitch of (2) was found to be about a fifth above (1), 
that of (3) about a major seventh above (1), and that of (4) a little 
higher again, forming an imperfect octave with the principal tone. 
For the corresponding modes of a uniform perfectly flexible mem- 
brane vibrating in vacuo, the theoretical intervals are those 
represented by the ratios 1*34, 1*66, 1*83 respectively ^ 

The vibrations of soap films have been investigated by Melde -. 
The frequencies for surfaces of equal area in the form of the circle, 
the square and the equilateral triangle, were found to be as 

1 Phil. Mag,, vol. vn., p. 160, 1879. 

2 Pogg. Ann., 159, p. 275, 1876. Akustik, p. 131, 1883. 



213.J FORCED VIBRATIONS. 349 

1*000 : 1'049 : 1'175. In membranes of this kind the tension is 
due to capillarity, and is independent of the thickness of the film.] 

213 a. The forced vibrations of square and circular membranes 
have been further experimentally studied by Elsas*, who has 
confirmed the conclusions of Savart as to the responsiveness of a 
membrane to sounds of arbitrary pitch. In these experiments the 
vibrations of a fork were communicated to the membrane by means 
of a light thread, attached normally at the centre ; and the position 
of the nodal curves and of the maxima of disturbance was traced 
in the usual manner by sand and lycopodium. A series of figures 
accompanies the memoir, shewing the eflFect of sounds of pro- 
gressively rising pitch. 

In many instances the curves found do not exhibit the 
symmetries demanded by the supposed conditions. Thus in 
the case of the square membrane all the curves should be similarly 
related to the four comers, and in the case of the circular mem- 
brane all the curves should be circles. The explanation is probably 
to be sought in the diflSculty of attaining equality of tension. If 
there be any irregularity, the eflFect will be to introduce modes of 
vibration which should not appear, as having nodes at the point of 
excitation, and this especially when there is a near agreement of 
periods. Or again, an irregularity may operate to disturb the 
balance between two modes of theoretically identical pitch, which 
should be excited to the same degree. Indeed the passage through 
such a point of isochronism may be expected to be highly unstable 
in the absence of moderate dissipative forces. 

The theoretical solution of these questions has already (§§ 196, 
204) been given, but would need much further development for 
an accurate determination of the nodal curves relating to periods 
not included among the natural periods. But the general course 
of the phenomenon can be traced without diflSculty. 

If the imposed frequency be less than the lowest natural 
frequency, the vibration is devoid of (internal) nodes. For a nodal 
line, if it existed, being of necessity either endless or terminated 
at the boundary*, would divide the membrane into two parts. Of 

^ Nova Acta der Ksl, Leap. Carol, DeuUchen Akademie, Bd. xly. Nr. 1. Halle, 
1S82. 

^ Otherwise the extremity would have to remain at rest under the action of 
component tensions from the surrounding parts which are aU in one direction. 



350 VIBRATIONS OF MEMBRANES. [213 a, 

these one part would be vibrating freely with a frequency less 
than the lowest natural to the whole membrane, an impossible 
condition of things (§ 211). The absence of nodal curves under the 
above-mentioned conditions is one of the conclusions drawn by 
Elsas from his observations. 

As the frequency of the imposed vibration rises through the 
lowest natural frequency, a nodal curve manifests itself round the 
point of excitation, and gradually extends. The course of things 
is most easily followed in the case of the circular membrane, 
excited at the centre. The nodal curves are then of necessity also 
circles, and it is evident that the first appearance of a nodal circle 
can take place only at the centre. Otherwise there would be a 
circular annulus of finite internal diameter, vibrating finely with a 
frequency only infinitesimally higher than that of the entire circle. 
At first sight indeed it might appear that even an infinitely small 
nodal circle would entail a finite elevation of pitch, but a con- 
sideration of the solution (§ 204) as expressed by a combination of 
BesseVs functions of the first and second kinds, shews that this is 
not the case. At the point of isochronism the second function 
disappears, and immediately afterwards re-enters with an infinitely 
small coefficient. But inasmuch as this function is itself infinite 
when r = 0, a nodal circle of vanishing radius is possible. Accord- 
ingly the fixation of the centre of a vibi-ating circular membrane 
does not alter the pitch, a conclusion which may be extended to 
the fixation of any number of detached points of a membrane of 
any shape. 

The eflFect of gradually increasing frequency upon the nodal 
system of a circular membrane may be thus summarized. Below 
the first proper tone there is no internal node. As this point is 
reached, the mode of vibration identifies itself with the corre- 
sponding free mode, and then an infinitely small nodal circle 
manifests itself. As the frequency further increases, this circle 
expands, until when the second proper tone is reached, it coincides 
with the nodal circle of the free vibration of this frequency. 
Another infinitely small circle now appears, and it, as well as the 
first, continually expands, until they coincide with the nodal system 
of a free vibration in the third proper tone. This process con- 
tinues as the pitch rises, every circle moving continually outwards. 
At each coincidence with a natural frequency the nodal system 
identifies itself with that of the free vibration, and a new circle 
begins to form itself at the centre. 



213 a.] 



NODAL CURVES. 



351 



The behaviour of a square membrane is of course more difficult 
to follow in detail. The transition from Fig. (34) case (4), corre- 
sponding to 7ti = 3, 71 = 1, and w = 1, n = 3, to Fig. (36) where m = 3, 
n = 3, can be traced in Elsas's curves through such forms as 

Fig. 39 a. 






CHAPTER X. 



VIBRATIONS OF PLATES \ 



214. In order to form axx^ording to Green's method the equa- 
tions of equilibrium and motion for a thin solid plate of uniform 
isotropic material and constant thickness, we require the expression 
for the potential energy of bending. It is easy to see that for each 
unit of area the potential energy F is a positive homogeneous 
symmetrical quadratic function of the two principal curvatures. 
Thus, if pi, p^ be the principal radii of curvature, the expression 
for V will be 



Aa^\^^) (1). 



where A and fi are constants, of which A must be positive, and 
fi must be numerically less than unity. Moreover if the material 
be of such a character that it undergoes no lateral contraction 
when a bar is pulled out, the constant /j, must vanish. This 
amount of information is almost all that is required for our 
purpose, and we may therefore content ourselves with a mere 
statement of the relations of the constants in (1) with those by 
means of which the elastic properties of bodies are usually de- 
fined. 

From Thomson and Tait s Natural Philosophy, §§ 639, 642, 
720, it appears that, if 2A be the thickness, q Young's modulus, 

^ [This Chapter deals only with jUxural vibrations. The extensional vibrations 
of an infinite plane plate are briefly considered in Chapter X.a, as a particolar 
case of those of an infinite cylindrical shell. They are not of much acoustical 
importance.] 



214.] POTENTIAL ENERGY OF BENDING. 358 

and fi the ratio of lateral contraction to longitudinal elongation 
when a bar is pulled out, the expression for V is 



3 (1 - /i») Wpi pj pipt 



• ••••••••• \st J • 



[Equation (2) gives the interpretation of the constants of (1) 
in its application to a homogeneous plate of isotropic material ; 
but the expression (1) itself is of far wider scope. The material 
composing the plate may vary from layer to layer, and the elastic 
character of any layer need not be isotropic, but only symmetrical 
with respect to the normal. As a particular case, the middle 
layer, or indeed any other layer, may be supposed to be physically 
inextensible. 

Similar remarks apply to the investigations of the following 
chapter relating to curved shells.] 

If w be the small displacement perpendicular to the plane 
of the plate at the point whose rectangular coordinates in the 
plane of the plate are x, y, 

pi 9%" ^' pi pi dx" df [dxdyj ' 
and thus for a unit of area, we have 

''-3(^,[('->-<'-'')|S^-(^)'}]«. 

which quantity has to be integrated over the sur&ce (8) of the 
plate. 

^ The foUowisg comparison of the notations need by the principal writers may 
save trouble to those who wish to consult the original memoirs. 

Bigidity=:n (Thomson) =ai (Lam6). 

Young's modulus =£ (Clebsch)=Jf (Thomson) = ^^- - (Thomson) 

^ fi(8w-n) /Thomson) =g (Kirchhoff and Donkin)=2JS:ii^ (Kirchhoff). 

in l + ff' _. 

Batio of lateral contraction to longitudinal elongation =^1 (Clebsch and Donkin) 

tn n. 6 \ 

= «r (Thomson)=-2^ (Thomson) = j-j-^-^ (Kirchhofl)=2— ^ (Lamfi). 
Poisson assumed this ratio to be i, and Wertheim \. 

R. 23 



» ■' L, 



A 



354 VIBRATIONS OF PLATES. [215. 

216. We proceed to find the variation of V, but it should be 

previously noticed that the second term in F, namely 1 1 — , 

represents the total curvature of the plate, and is therefore de- 
pendent only on the state of things at the edge. 

SO that we have to consider the two variations 

jjV^w.V^Sw.dS and ||s (/h />,)-' dS. 
Now by Green's theorem 
(h^w . V^Sw . d8 = jh^w .Bw.dS 

-j-^T '^^'ds + jV^w-^ds (2), 

in which da denotes an element of the boundary, and d/dn denotes 
differentiation with respect to the normal of the boundary drawn 
outwards. 

The transformation of the second part is more difficult. We 

have 

IcPw cPSw d^w d^Sw __ g d^w d^Sw ) ,^ 



C-//{: 



da^ dy* dy* da^ dxdy dxdy] 

The quantity under the sign of integration may be put into 
the form 

d^ /dBw d^w _ dSw dhv\ jd / dSw d^w _ dSw d"w \ 
dy \ dy da^ dx dxdy) dx\dx dy^ dy dxdy) ' 

Now, if i^ be any function of x and y, 

ll--j-dxdy^ iFBinOds 

rr^p r } (3)> 

If ^ da?dy =/ i^ cos ^ dfi 

where is the angle between x and the normal drawn outwards, 
and the integration on the right-hand side extends round the 
boundary. Using these, we find 

S [f — = (ds sin {— ^ « ^^ ^^ \ 
J J PiRi J [dy daf^ dx dxdy] 



4- (d ff J^^ ^^ — ^^^ ^*^^ I 
J \dx dy^ dy dxdy] ' 



215.] 



EXPRESSION FOR 87. 



355 



If we substitute fgr dZwjdx, dSw/dy their values in terms 
dSw/dn, dSw/ds, from the equations (see Fig. 40) 



dSw dhw ^ dSw . ^ 
QQQff — ;— sin^ 



dx dn 



da 



dhw dSw . >, . d6w A 
j— = -j — sm^+ J— cos^ 
dy dn da 

Fig. 40. 



(4), 




we obtain 



— = \da-j-<sm^6-j-z+co8^6 :j-r-2smdcos^ ,-^-l 

The second integral by a partial integration with respect to 
a may be put into the form 

C!ol]ecting and rearranging our results, we find 



-I 



Swda- 



(d^'^w 



dn 



+ (l-A^)5^(cos^sm^(^-^j 



f ^ ^ 1^^'^ + (1 - /^) (cos' e 



+ 2co8^8in^^^)i ...(6). 

23—2 



356 VIBRATIONS OF PLATES. [215, 

There will now be no difficulty in forming the equations of 
motion. If p be the volume density, and Z.p.2h.dS the transverse 
force acting on the element dS, 

SV-jf2ZphBwd8+jJ2phwSwdS^0 (7y 

is the general variational equation, which must be true whatever 
function (consistent with the constitution of the system) Sw may 
be supposed to be. Hence by the principles of the Calculus of 
Variations 

g^^jV'«,-Z+«=0 (8). 

at every point of the plate. 

If the edges of the plate be free, there is no restriction on the 
hypothetical boundary values of Bw and dSw/dn, and therefore the 
coefficients of these quantities in the expression for S Fmust vanish. 
The conditions to be satisfied at a free edge ai*e thus 



dn 



+ 



+ (co8.<?-8in«^)|Jy-}=0 



.(9). 



4- 2co8^sin ^ T-j-h =0 

dxdy) 

If the whole circumference of the plate be clamped, Bw^O, 
d&w/dn = 0, and the satis&ction of the boundary conditions is 
already secured. If the edge be ' supported'*, Sw — 0, but dSw/dn 
is arbitrary. The second of the equations (9) must in this case be 
satisfied by w. 

216. The boundary equations may be simplified by getting 
rid of the extrinsic element involved in the use of Cartesian 
coordinates Taking the axis of x parallel to the normal of the 
bounding curve, we see that we may write 

cos* j-T + sm* ^ J-, + 2 cos tf sm j— r- = 3- - . 
cur" ay" dxdy dn^ 

AUo V.«;=^-+^, (1). 

^ The rotatory inertia is here neglected. > Compare § 162. 



216.] CONDITIONS FOR A FREE EDGE. 357 

where <r is a fixed axis coinciding with the tangent at the point 
under consideration. In general cPw/da^ differs from (Pw/dsK 
To obtain the relation between them, we may proceed thus. 
Expand w by Maclaurin's theorem in ascending powers of the 
small quantities n and c, and substitute for n and a their values 
in terms of 8, the arc of the curve. 

Thus in general 

while on the curve cr = « + cubes, n = — ^ s^jp + . .. , where p is 
the radius of curvature. Accordingly for points on the curve, 

.dws^dw . d?w . , , - 

ti; = Wo — Aj hj—^ + i J— J ^ + cubes of «, 

^ dno p ao-o * a<7o* 



^^0 



, . , - c^w d^w 1 dw .^. 

and therefore -ri = j-t j- (2); 

d^ da^ p dn ^ ^ * 

whence from (1) 

-_ d^w Idw ^ d^w ,-. 

^'"-'M-^-pdi.-^l^ <3>- 

We conclude that the second boundary condition in (9) § 215 
may be put into the form 

d?w ^ (IdAi) d?w\ . 

di^^-^^y^dn-^-dFr^ (*>• 

In the same way by putting ^ = 0, we see that 

is equivalent to d^w/dndc, where it is to be understood that 
the axes of n and <r are fixed. The first boundary condition now 
becomes 

|v.„.a-rt|(^^)=o (.> 

If we apply these equations to the rectangle whose sides are 
parallel to the coordinate axes, we obtain as the conditions to be 
satisfied along the edges parallel to y, 

d fd^w d^w) ^y 

^{d^-^(2-;.)^.}=a^ ^^^ 

d^w d?w ^ 



358 VIBRATIONS OF PLATBa [216. 

In this case the distinction between a and a disappears, and p^ the 
radius of curvature, is infinitely great. The conditions for the 
other pair of edges are found by interchanging x and y. These 
results may be obtained equally well from (9) § 215 directly, with- 
out the preliminary transformation. 

217. If we suppose Z = 0, and write 

zfii^r"^ <^>' 

the general equation becomes 

t£; + c*V*w = (2), 

or, if w X cos {pt — €), 

V*w = k'w (3), 

where k^=^p^/c^ (4). 

Any two values of te;, u and v, corresponding to the same 
boundary conditions, are conjugcUe, that is to say 



// 



uvdS^O (5), 

provided that the periods be different In order to prove this 
from the ordinary differential equation (3), we should have to 
retrace the steps by which (3) was obtained. This is the method 
adopted by Kirchhoff for the circular disc, but it is much simpler 
and more direct to use the variational equation 



BV+2phJlw&wdS=^0 (6), 



in which w refers to the actual motion, and Su; to an arbitrary 
displacement consistent with the nature of the system. SFis a 
symmetrical function of w and Bw, as may be seen ftom § 215, or 
from the general character of V (§ 94). 

If we now suppose in the first place that w^u, Sw = v, we 
have 



&r=2php'jjuvdS; 



and in like manner if we put w = v, Sw^u, which we are equally 
entitled to do. 



SF=: 2php'*jjuvdS, 



217.} OONJUGATB PROPERTY. 359 



whence 



(P^^p'')jjuvd8^0 .:'. (7). 



This demonstration is valid whatever may be the form of the 
boundary, and whether the edge be clamped, supported, or firee, in 
whole or in part 

As for the case of membranes in the last Chapter, equation 
(7) may be employed to prove that the admissible values of p^ are 
real ; but this is evident from physical considerations. 

218. For the application to a circular disc, it is necessary to 
express the equations by means of polar coordinates. Taking the 
centre of the disc as pole, we have for the general equation to be 
satisfied at all points of the area 

(V*-&*)t(;»0 (1), 

wl,e«(§200, V..*+i4 + l*. 

To express the boundary condition (§ 216) for a free edge 
(r = a), we have 

d — , _ ^ rra d / d^w \ __ d d^ / dw\ d^w _ d?w 
dfi ^^Tr ^' diKdliid^J^^Ldech-K^)' (&»~^5^' 

p = radius of curvature ^ a ; and thus 

dr\df^'^ rdr)'^de»\ a» dr a* ^)^ ^ .„. 

c \^)' 

d?w (1 dw 1 d^w\ _ ^ 

After the differentiations are performed, r is to be made equal 
to a. 

If whe expanded in Fourier's series 

each term separately must satisfy (2), and thus, since 

Wn « cos (n0 — a), 

d fd^Wnldw^\ fi-fidwn 3-A*.„ '\_ A 
Tr\d?~'^rW)^''\lir dr'' a» "^V"". ^g^ 

d^Wn . /I dWn W* \ A 



360 VIBRATIONS OF PLATES. [218. 

The superficial differential equation may be written 

(V« + ifc») ( V« - ifc») w = 0, 
which becomes for the general term of the Fourier expansion 

shewing that the complete value of Wn will be obtained by adding 
together, with arbitrary constants prefixed, the general solutions of 

■ it-lTr-t^^h-" <*^ 

The equation with the upper sign is the same as that which 
obtains in the case of the vibrations of circular membranes, and 
as in the last Chapter we conclude that the solution applicable 
to the problem in hand is Wn « Jn (At), the second function of r 
being here inadmissible. 

In the same way the solution of the equation with the lower 
sign iawnocjn (ikr), where i = V (— 1) as usual. [See § 221 a.] 
The simple vibration is thus 
Wn = cos nO {aJn (At) + fiJn (ikr)} + sin nd [yJn (kr) + BJn (ikr)]. 

The two boundary equations will determine the admissible 
values of k and the values which must be given to the ratios 
a : fi and y : S. From the form of these equations it is evident 
that we must have a : fi = y : S, 

and thus Wn may be expressed in the form 

Wn = P cos (nd — a) {/„ (At) + X«/„ (ikr)] cos (pt — e) (5). 

As in the case of a membrane the nodal system is composed of 
the n diameters symmetrically distributed round the centre, but 
otherwise arbitrary, denoted by 

cos(n5-a) = (6), 

together with the concentric circles, whose equation is 

Jn(kr) + \Jn(ikr) = (7). 

219. In order to determine \ and k we must introduce the 
boundary conditions. When the edge is free, we obtain from 
(3) § 218 

n^(M"'l)[kaJn(ka)-Jn{ka)}-k^a*Jn'(ka) \ 
n V - 1) {ikaJn'iika) - Jn{ika)} + ik'a'Jn' (ika) 

_ (fjL'-l){kaJn'(ka) -n^Jn{ka)] - k'a ^Jn (ka) 1 

(ji - l){%kaJn' (ika)- n^Jn (tfca)}"+ k^a'Jn (ika) ) 



...(1), 



219.] POLAR CO-ORDINATBS. 361 

in which use has been made of the di£ferential equations satisfied 
by JnQcr\ Jn{ilcr), In each of the fractions on the right the 
denominator may be derived from the numerator by writing ik in 
place of k. By elimination of X the equation is obtained whose 
roots give the admissible values of h 

When n =s 0, the result assumes a simple form, viz. 

This, of course, could have been more easily obtained by neglecting 
n from the beginning. 

The calculation of the lowest root for each value of n is trouble- 
some, and in the absence of appropriate tables must be effected 
by means of the ascending series for the functions Jn (kr), Jn (ikr). 
In the case of the higher roots recourse may be had to the semi- 
convergent descending series for the same functions. Earchhoff 
finds 

tan(to-i/i7r)- -i::i:i/(8ifca)i:^/(8ika)> + . . . ^^^' 

where 

5 = 7(l-4n»)-8, 
C=7(l-4n«)(9-4n«) + 48(lH-4n«), 

2) = - Yi {(1 - 4^^") (9 - *w') (13 - ^.n^} + 8 (9 + 136n« + SOn*). 

When ka is great, 

tan {ka - \ rnr) = approx. ; 
whence 

A;a = i7r(n + 2A) (4), 

where h is an integer. 

It appears by a numerical comparison that h is identical with 
the number of circular nodes, and (4) expresses a law discovered 
by Chladni, that the frequencies corresponding to figures with a 
given number of nodal diameters are, with the exception of the 
lowest, approximately proportional to the squares of consecutive 
even or uneven numbers, according as the number of the diameters 
is itself even or odd. Within the limits of application of (4), we 
see also that the pitch is approximately unaltered, when any 
number is subtracted frt)m h, provided twice that number be 



362 



VIBRATIONS OF PLATES. 



[219. 



added to ;i. This law, of which traces appear in the following table, 
may be expressed by saying that towards raising the pitch nodal 
circles have twice the effect of nodal diameters. It is probable, 
however, that, strictly speaking, no two normal components have 
exactly the same pitch. 



h 



1 

2 
h 


1 


n = 


n=l 


Ch. 

• • • 

Ois 
gi««' + 


P. 

• ■ • 

Ois + 
b'- 


W. 

• • • 

A + 
b' + 


Ch. 

• . • 

b 

e" + 


P. 

. • • 

h- 


W. 

• ■ • 

c- 

fi8" + 


n=2 


n=3 


Ch. 
C 

g' 


P. 
C 

gi8' + 


W. 
C 
a'- 


Ch, 
d 
d".difl" 


p. 

dis — 
di8" + 


w. 

dis — 
e"- 



The table, extracted from Kirchhoff's memoir, gives the pitch 
of the more important overtones of a free circular plate, the gravest 
being assumed to be (7 ^ The three columns under the heads 
Ch, P, W refer respectively to the results as observed by Chladni 
and as calculated from theory with Poisson's and Wertheim's 
values of fi, A pliLS sign denotes that the actual pitch is a little 
higher, and a minus sign that it is a little lower, than that written. 
The discrepancies between theory and observation are considerable, 
but perhaps not greater than may be attributed to irregularity in 
the plate. 

220. The radii of the nodal circles in the sjrmmetrical case 
(n = 0) were calculated by Poisson, and compared by him with 
results obtained experimentally by Savart. The following numbers 
are taken from a paper by Strehlke^ who made some careful 
measurements. The radius of the disc is taken as unity. 

Obfleryation. Cftleolation. 

One circle ... 0-67815 0-68062. 



Two circles, 



fO-39133 
'•••(0-84149 

[0-25631 
Three circles i 0-59107 

lo-89360 



0-39151. 
0*84200. 

0-25679. 
0-59147. 
0-89381. 



^ Gis corresponds to Gj( of the English notation, and htob natural. 

s Fogg. Ann. xcr. p. 577. 1855. 



220,] 



kirchhopf's theory. 



363 



The calculated results appear to refer to Poisson's value of fi, but 
would vary very little if Wertheim's value were substituted. 

The following table gives a comparison of EarchhofiTs theory 
(n not zero) with measurements by Strehlke made on less accurate 
discs. 

Radix of Circular Nodes. 



n 
n 
n 



1, h 

2, h 

3, h 



1 
1 
1 



n = l, A = 2 



Observation. 


Calculation. 




^ = }(P.). ,t = i(W.). 


0-781 0-783 0-781 0-783 


0-78136 


0-78088 


0-79 0-81 0-82 


082194 


0-82274 


0-838 0-842 


0-84523 


0-84681 


0-488 0-492 


0-49774 


0-49715 


0-869 0-869 


0-87057 


0-87015 



The most general motion of the uniform circular plate is 
expressed by the superposition, with arbitrary amplitudes and 
phases, of the normal components already investigated. The 
determination of the amplitude and phase to correspond to 
arbitrary initial displacements and velocities is effected precisely 
as in the corresponding problem for the membrane by the aid of 
the characteristic property of the normal functions proved in § 217. 

221. When the plate is truly symmetrical, whether uniform 
or not, theory indicates, and experiment verifies, that the position 
of the nodal diameters is arbitrary, or rather dependent only on 
the manner in which the plate is supported, and excited. By 
varying the place of support, any desired diameter may be made 
nodal It is generally otherwise when there is any sensible 
departure from exact sjrmmetry. The two modes of vibration, 
which originally, in consequence of the equality of periods, could 
be combined in any proportion without ceasing to be simple 
harmonic, are now separated and affected with different periods. 
At the same time the position of the nodal diameters becomes 
determinate, or rather limited to two alternatives. The one set is 
derived frt)m the other by rotation through half the angle included 
between two adjacent diameters of the same set. This supposes 
that the deviation from uniformity is small ; otherwise the nodal 
system will no longer be composed of approximate circles and 
diameters at all The cause of the deviation may be an irregu- 
larity either in the material or in the thickness or in the form of 



364 VIBRATIONS OF PLATES. [221, 

the boundary. The effect of a small load at any point may be 
investigated as in the parallel problem of the membrane § 208. 
If the place at which the load is attached does not lie on a nodal 
circle, the normal types are made determinate. The diametral 
system corresponding to one of the types passes through the place 
in question, and for this type the period is unaltered. The period 
of the other type is increased. 

[The divergence of free periods, which is due to slight in- 
equalities, would seem to afford an explanation of some curious 
observations by Savart^ When a circular plate, vibrating with 
nodal diameters, is under the influence of the bow applied at any 
part of the circumference, the nodal diameters indicated by sand are 
so situated that the bow lies in the middle of a vibrating segment. 
If, however, the bow be suddenly withdrawn, the nodal system 
oscillates, or even revolves, during the subsidence of the motion. 
It is evident that no such displacement could be expected, 
were the plate absolutely symmetrical. The same would be true, 
even in the case of asymmetry, if the bow were so applied as to 
excite one only of the two determinate vibmtions then possible. 
But in general the effect of the bow must be to excite both kinds 
of vibrations, and then the matter is more complicated. It would 
seem that so long as the constraining action of the bow lasts, both 
vibrations are forced to keep the same time, and the effect is 
much the same as in the case of symmetry. But on withdrawal 
of the bow the free vibrations which then ensue take place each in 
its proper frequency, and a phase difference soon arises by which 
the effects are modified. 

Let us suppose that the origin of is so chosen in relation 
to the irregularities that the types of vibration are represented 
by cos nO, sin nO. Then in general the free vibrations, resulting 
frx>m the action of the bow at an arbitrary point of the circum- 
ference, may be taken to be 

cosna sin n^ cos pi — sin na cos n^ cos (pi + e) (1), 

where e is the difference of phase which has accumulated since 
the commencement of the free vibrations. In the case of 
sjrmmetry 6 = 0, and (1) becomes 

sin n(^ — a) cos pi (2), 

1 Ann. Chim., vol. 86, p. 267, 1827. 



221.] OSCILLATION OF NODES. 365 

which represents a fixed nodal system 

e^a + miir/n) (3), 

in any arbitrary position depending upon the point of application 
of the bow. A similar fixity of the nodal system occurs, in spite 
of the variable €, when a is so chosen that cos na «> or sin na == 0. 
But in general there is no fixed nodal system. When € is a 
multiple of 27r, that is when the two vibrations are restored to 
the same phase, there is a nodal system represented by (3). And 
when € is an odd multiple of tt, so that the two vibrations are in 
opposite phases, we have in place of (2) 

8inw(^ + a)cosp^ (4), 

with a nodal system 

5 = -a-f m(7r/n) (5). 

In these cases there is a nodal system, and in a sense the system 
may be said to oscillate between the positions given by (3) and (5) ; 
but it must not be overlooked that at intermediate times there is 
no true nodal system at all. Thus, when e = Jtt, (1) becomes 

cos na sin n0 cospt + sin na cos nO sin pt 

The squared amplitude of this motion is 

cos' na sin' nd + sin' na cos* nO, 

a quantity which does not vanish for any value of 0. In general 
the squared amplitude is 

cos' na sin' n6 -h sin' na cos' nd -2 cos na sin na cos nO sin nO cos e, 

or, as it may also be written, 

^ — ^cos2na cos 2n0 — ^sin2na sin 2n0 cose (6). 

This quantity is a maximum or a minimum when 

tan 2ndsscos€ tan2na (7). 

The minimum of motion thus oscillates backwards and forwards 
between tf = + a and ^ = — a ; but as we have seen, it is only in 
these extreme positions that the minimum is zero. 

A like phenomenon occurs during the free vibrations of a 
circular membrane, or in fact of any system of revolution such 
that the position of nodal lines is arbitrary so long as the 
symmetry is complete.] 



366 VIBRATIONS OF PLATES. [221. 

The two other cases of a circular plate in which the edge 
is either clamped or siipported would be easier than the preceding 
in their theoretical treatment, but are of less practical interest on 
account of the difficulty of experimentally realising the conditions 
assumed. The general result that the nodal qrstem is^compoeed 
of concentric circles, and diameters symmetrically distributed, is 
applicable to all the three cases. 

221a. The use in the telephone of a thin circular plate 
clamped at the edge lends a certain interest to the calculation of 
the periods and modes of vibration of such a plate. It will suffice 
to consider the sjrmmetrical modes. 

By (5) § 218 we may take as representing the motion in 

this case 

w^Jo(kr)'¥\Jo{ikr)^Jo{kr)'\-\Io{kr) (1), 

from which 

^ = J-;(Jtr) + i\Jo'(»AT) = -/,(AT) + X/x(AT) (2), 

where we write 

/o(^) = /o(i«) = l+| + 2i^ + (3), 

Ii (^) ^iJo{iz) = 2 + 2^ + 2M^ "^ (*^- 

Since the plate is clamped at r = a, both w and dw/dr must 
there vanish. Hence, writing ka^z, we get as the frequency 
equation 

Mz)'^i,(zr^ (^>- 

In (5) /i and Iq are both positive, so that the signs of Ji and J^ 
must be opposite. Hence by Table B § 206 the first root must 
lie between 2*4 and 3*8, the second between 5*5 and 7*0, and 
so on. The values of the earlier roots might be obtained without 
much difficulty from the series for /o and Ii by using the table 
§ 200 for t/o Ai^d t/i ; but it will be convenient for the present and 
further purposes to give a short table ^ of the functions /« and /^ 
themselvea For large values of the argument descending series, 
analogous to (10) § 200, may be employed. 

^ Caloolated by A. Lodge, Brit. An, Rep.y 1889. 



221a.] 



CLAMPED BOUyDABT. 



367 



t 


JoW 


1 


M 


^oW 


JiW 


oo 


1-0000 


0-0000 


3-0 


4-8808 


3-9534 


•2 


1-0100 


•1005 


3*2 


5*7472 


4-7343 


•4 


1-0404 


•2040 


3*4 


6*7848 


5-6701 


•6 


1-0920 


•3137 


3*6 


8-0277 


6-7927 


•8 


11665 


•4329 


3*8 


9*5169 


8-1404 


10 


1-2661 


5652 


4-0 


11*3019 


9-7595 


1-2 


1-3937 


7147 


4*2 


13*4425 


11^7056 


1-4 


1*5534 


*8861 , 


4*4 


16-0104 


14-0462 


1-6 


1-7500 


1-0848 1 


4-6 


19-0926 


16*8626 


1-8 


1-9896 


1-3172 


4-8 


22-7937 


20*2528 


20 


2-2796 


1-5906 


5-0 


27-2399 


24-3356 


2*2 


2*6291 


1-9141 1 


5-2 


32-5836 


29-2543 


2*4 


3-0493 


2-2981 


5-4 


39-0088 


35*1821 


2-6 


3-5533 


2*7554 


5*6 


46-7376 


42-3283 


2-8 


4-1573 


33011 


5*8 


56-0381 


50-9462 






, 


6-0 


67-2344 


61-3419 



The first root of (5) is ^ = 3*20. This then is the value of ka 
for the gravest sjrmmetrical vibration. The next value oi z is 
about 6'3. Since the frequency varies as i* (§ 217), the interval 
between the tones is nearly two octaves. 

Returning to the first root, we have for the fi^uency (n) 

§217, 

_ p _ (3'2)«c' _ (3-2) Vg . A 






(6). 



27r 2wa« 27ra« V3/» (i - /a«J 

This is the general formula. For rough calculations fi* in the 
denominator may be omitted. If for the case of iron we take 

p ^ 7-7, g = 20 X 10" 
2-4 X 10».2A 



we find 



n = 



a' 



(7), 



2h and a being expressed in centimetres. 

A telephone plate measured by the author gave 

a = 2-2, 2A = 020. 
According to these values 

n =s 991 vibrations per second. 

222. We have seen that in general Chladni's figures as traced 
by sand agree very closely with the circles and diameters of 
theory; but in certain cases deviations occur, which are usually 
attributed to irregularities in the plate. It must however be re- 



368 VIBRATIONS OF PLATES. [222. 

membered that the vibrations excited by a bow are not strictly 
speaking free, and that their periods are therefore liable to a 
certain modification. It may be that under the action of the bow 
two or more normal component vibrations coexist. The whole 
motion may be simple harmonic in virtue of the external force, 
although the natural periods would be a little different. Such an 
explanation is suggested by the regular character of the figures 
obtained in certain cases. 

Another cause of deviation may perhaps be found in the 
manner in which the plates are supported. The requirements of 
theory are often difficult to meet in actual experiment When 
this is so, we may have to be content with an imperfect compari- 
son ; but we must remember that a discrepancy may be the fault 
of the experiment as well as of the theory. 

[In the ordinary use of sand to investigate the vibrations of 
flat plates and membranes the movement to the nodes is irregular 
in its character. If a grain be situated elsewhere than at a node, 
it is made to jump by a sufficiently vigorous transverse vibration. 
The result may be a movement either towards or from a node ; 
but after a succession of such jumps the grain ultimately finds its 
way to a node as the only place where it can remain undisturbed. 
Grains which have already arrived at a node remain there, while 
others are constantly shifting their position. 

It was found by Savart that very fine powder, such as lyco- 
podium, behaves differently from sand. Instead of collecting at 
the nodes, it heaps itself up at the places of greatest motion. 
This effect was traced by Faraday* to the influence of currents of 
air, themselves the result of the vibration. In a vacuum all 
powders move to the nodes. 

In some cases the movement of sand to the nodes, or to some 
of them, takes place in a more direct manner as the result of 
friction. Thus, in his investigation of the longitudinal vibrations 
of thin narrow strips of glass, held horizontally, Savart* observed 
the delineation of nodes apparently dependent upon an accom- 
paniment of vibrations of a transverse character. The special 
peculiarity of this phenomenon was the non-correspondence of the 
lines traced by sand upon the two faces of the glass when tested 

1 On a Peoaliar Class of Aoonstical Figures, Phil, Trans., 1S81, p. 299. 
< Am. d. Chim,, vol. U, p. 113, 1S20. 



222.] SAV art's observations. 369 

in succession, a fact sufficient to shew that the transverse motion 
was connected with a failure of uniformity. In consequence of 
this there are developed transverse vibrations of the same (high) 
pitch as that of the principal longitudinal motion, and therefore 
attended with many nodes. These nodes are of course the same 
whichever face of the glass is uppermost, and it might be supposed 
that they would all be indicated by the sand, as would happen if 
the transverse vibrations existed alone. But the combination of 
the two kinds of motion causes a creeping of the sand towards the 
alternate nodes, the movements of the sand at corresponding 
points on the two sides of the plate being always in opposite 
directions. On the one side an inwards longitudinal motion (for 
example) is attended by an upwards transverse motion, but when 
the plate is reversed the same inwards longitudinal motion is 
associated with a transverse motion directed downwards. If there 
were no transverse motion, the longitudinal force upon any 
particle resulting from friction would vanish in the long run, but 
in consequence of the transverse motion this balance is upset, and 
in a manner different upon the two sides of the plate. The above 
considerations appear to afford sufficient ground for an explanation 
of the remarkable phenomenon observed by Savart, but an attempt 
to follow the matter further into detail would lead us too 
far\] 

223. The first attempt to solve the problem with which we 
have just been occupied is due to Sophie Germain, who succeeded 
in obtaining the correct differential equation, but was led to 
erroneous boundary conditions. For a free plate the latter part of 
the problem is indeed of considerable difficulty. In Poisson's 
memoir *Sur T^quilibre et le mouvement des corps flastiquesV 
that eminent mathematician gave three equations as necessary to be 
satisfied at all points of a free edge, but Eirchhoff has proved that 
in general it would be impossible to satisfy them alL It happens, 
however, that an exception occurs in the case of the symmetrical 
vibrations of a circular plate, when one of the equations is true 
identically. Owing to this peculiarity, Poisson's theory of the 
symmetrical vibrations is correct, notwithstanding the error in his 
view as to the boundary conditions. In 1850 the subject was 



^ See Terqnem, C. 12., zlyx., p. 775, 1858. 
2 Mim. de VAcad. d. Sc. b, Par, 1829. 

B. 24 



370 VIBRATIONS OF PLATES, [223. 

resumed by Eirchhoff ^ who first gave the two equations appropriate 
to a free edge, and completed the theory of the vibrations of a 
circular disc. 

224. The correctness of Earchhoff's boundary equations has 
been disputed by Mathieu', who, without explaining where he 
considers EirchhofiTs error to lie, has substituted a different set of 
equations. He proves that if u and u be two normal functions, so 
that w^ucospt, w=iu' cos p't are possible vibrations, then 



(p^ - p'^) j j uu' dxdy 



^c^ I d8\u 



,d^^u -, du' -, ,du dVW 
an an an an 



,(1). 



This follows, if it be admitted that u, v! satisfy respectively 
the equations 

c* V^u^p^u, c* V*i£' « p'^u\ 

Since the left-hand member is zero, the same must be true of 
the right-hand member; and this, according to Mathieu, cannot 
be the case, unless at all points of the boundary both u and u' 
satisfy one of the four following pairs of equations : 



u =0 I V«u =0 



u =0 1- =0 



dn 



an J dn J J dn 



J 



Y ' 



The second pair would seem the most likely for a free edge, but 
it is found to lead to an impossibility. Since the first and third 
pairs are obviously inadmissible, Mathieu concludes that the fourth 
pair of equations must be those which really express the condition 
of a free edge. In his belief in this result he is not shaken by the 
fisict that the corresponding conditions for the free end of a bar 
would be du/dx = 0, d^u/da^ = 0, the first of which is contradicted 
by the roughest observation of the vibration of a large tuning- 
fork. 



^ Crelle, t. zl. p. 51. Ueber das Oleichgewicht und die Bewegnng einer elas- 
tischen Soheibe. 

> LiouviUe, t. ziv. 1869. 



224.] HISTOBY OF PROBLEM. 371 

The fact is that although any of the four pairs of equations 
would secure the evanescence of the boundary integral in (1), it 
does not follow conversely that the integral can be made to vanish 
in no other way; and such a conclusion is negatived by EarchhofiTs 
investigation. There are besides innumerable other cases in 
which the integral in question would vanish, all that is really 
necessary being that the boundary appliances should be either at 
rest, or devoid of inertia. 

226. The vibrations of a rectangular plate, whose edge is 
supported, may be easily investigated theoretically, the normal 
functions being identical with those applicable to a membrane of 
the same shape, whose boundary is fixed. If we assume 

w = sm- sm —j-^ cospt (1), 

we see that at all points of th^ boundary, 

w = 0, d'wld^=^0, d^w/df=^0, 

which secure the fulfilment of the necessary conditions (§ 215). 
The value of p, found by substitution in c^V*w^p^w, 



is 



^-^H?+S <2)' 



shewing that the analogy to the membrane does not extend to the 
sequence of tones. 

It is not necessary to repeat here the discussion of the primary 
and derived nodal systems given in Chapter ix. It is enough to 
observe that if two of the fundamental modes (1) have the same 
period in the case of the membrane, they must also have the same 
period in the case of the plate. The derived nodal systems are 
accordingly identical in the two cases. 

The generality of the value of w obtained by compounding 
with arbitrary amplitudes and phases all possible particular solu- 
tions of the form (1) requires no fresh discussion. 

Unless the contrary assertion had been made, it would have 
seemed unnecessary to say that the nodes of a supported plate 
have nothing to do with the ordinary Chladni's figures, which 
belong to a plate whose edges are free. 

24—2 



372 VIBRATIONS OF PLATES. [225. 

The realization of the conditions for a supported edge is 
scarcely attainable in practice. Appliances are required capable 
of holding the boundaiy of the plate at rest, and of such a nature 
that they give rise to no couples about tangential axes. We may 
conceive the plate to be held in its place by friction against the 
walls of a cylinder circumscribed closely round it. 

226. The problem of a rectangular plate, whose edges are 
free, is one of great diflSculty, and has for the most pcui; resisted 
attacks If we suppose that the displacement w is independent 
of y, the general differential equation becomes identical with that 
with which we were concerned in Chapter viil If we take the 
solution corresponding to the case of a bar whose ends are free, 
and therefore satisfying dhu/da^==0, dhu/da^^^O, when a? = and 
when a? = a, we obtain a value of w which satisfies the general 
differential equation, as well as the pair of boundary equations 






dx 



da^ dy\ 



= 



d^w d^w ^ 



\ (1). 



which are applicable to the edges parallel to y ; but the second 
boundary condition for the other pair of edges, namely 

^ + ^dii = ° <2). 

will be violated, unless fi^O. This shews that, except in the 
case reserved, it is not possible for a free rectangular plate to 
vibrate after the manner of a bar; unless indeed as an approxima- 
tion, when the length parallel to one pair of edges is so great that 
the conditions to be satisfied at the second pair of edges may be 
left out of account. 

Although the constant ^i (which expresses the ratio of lateral 
contraction to longitudinal extension when a bar is drawn out) 
is positive for every known substance, in the case of a few sub- 
stances — cork, for example — it is comparatively very small There 
is, so far as we know, nothing absurd in the idea of a substance 

^ [The case where two opposite edges are free while the other two edges are 
supported, has been discussed by Yoigt {Qdttingen Nachriekten, 1898).] 



226.] RECTANGULAR PLATE, 373 

for which fi vanishes The investigation of the problem under 
this condition is therefore not devoid of interest, though the results 
will not be strictly applicable to ordinary glass or metal plates, 
for which the value of /tt is about ^} 

liuiy ?^, &c. denote the normal functions for a free bar inves- 
tigated in Chapter viii., corresponding to 2, 3, nodes, the 

vibrations of a rectangular plate will be expressed by 

w^Ui {xla)y \o = Ua {xja), &c., 

or w^ Ui (y/b), xo=^ ^iiy/b), &c. 

In each of these primitive modes the nodal system is composed 
of straight lines parallel to one or other of the edges of the 
rectangle. When 6 = a, the rectangle becomes a square, and the 
vibrations 

w = tin {xja), w^Un (y/a), 

having necessarily the same period, may be combined in any pro- 
portion, while the whole motion still remains simple harmonia 
Whatever the proportion may be, the resulting nodal curve will of 
necessity pass through the points determined by 

Un (x/a) = 0, Un (y/a) = 0. 

Now let us consider more particularly the case of n = 1. The 
nodal system of the primitive mode, w = Uj (x/a), consists of a 
pair of straight lines parallel to y, whose distance from the nearest 
edge is '2242 a. The points in which these lines are met by the 
corresponding pair for w == v^ (2/l<^)» are those through which the 
nodal curve of the compound vibration must in all cases pasa It 
is evident that they are symmetrically disposed on the diagonals 
of the square. If the two primitive vibrations be taken equal, 
but in opposite phases (or, algebraically, with equal and opposite 
amplitudes), we have 

w= iti (xfa) - Ml (y/a) (3), 

^ In order to make a plate of material, for which fi is not zero, vibrate in the 
manner of a bar, it would be necessary to apply constraining couples to the edges 
parallel to the plane of bending to prevent the assumption of a contrary curvature. 
The efifect of these couples would be to raise the pitch, and therefore the calcu- 
lation founded on the type proper to fi=0 would give a result somewhat higher in 
pitch than the truth. 



374 



VIBRATIONS OF PLATES. 



[226. 




from which it is evident that w vanishes when a? = y, that is along 

the diagonal which passes through the origin. Fig. 41. 

That w will also vanish along the other diagonal 

follows from the symmetry of the functions, and 

we conclude that the nodal system of (3) comprises 

both the diagonals (Fig. 41). This is a well-known 

mode of vibration of a square plate. 

A second notable case is when the amplitudes are equal and 
their phases the same, so that 

w = u^{x/a)-^u,(y/a) (4). 

The most convenient method of constructing graphically 
the curves, for which w = const., is that employed by Maxwell 
in similar cases. The two systems of curves (in this instance 
straight lines) represented by i^i (x/a) = const., u^ (y/ct) = const., are 
first laid down, the values of the constants forming an arith- 
metical progression with the same common difference in the two 
cases. In this way a network is obtained which the required 
curves cross diagonally. The execution of the proposed plan 
requires an inversion of the table given in Chapter viii., § 178, 
expressing the march of the function Wj, of which the result is as 
follows : — 



w, 


X : a 


^1 


X : a 


+ 1-00 


•5000 


- -25 


•1871 


•75 


•3680 


•50 


•1518 


•50 


•3106 


•75 


•1179 


•25 


•2647 


1-00 


•0846 


•00 


•2242 


r25 


•0517 






-1^50 


•0190 



The system of lines represented by the above values of x (com- 
pleted symmetrically on the ftirther side of the central line) and 
the corresponding system for y are laid down in Fig. 42. From 
these the curves of equal displacement are deduced. At the 
centre of the square we have w a maximum and equal to 2 on the 
scale adopted. The first curve proceeding outwards is the locus of 
points at which ti; = 1. The next is the nodal line, separating the 
regions of opposite displacement. The remaining curves taken in 



226.] CASE OF A SQUABE FULTB. 375 

order give the displacements — 1, — 2, — 3. The numerically great* 
est negative displacement occurs at the comers of the aquare, 
where it amounts to 2 x 1-643 = 3-290.' 

The nodal curve thus constructed agrees pretty closely with the 
observations of Strehlke *. His results, which refer to three care- 
fully worked plates of glass, are embodied in the following polar 
equations : 

-40143 -01711 -00127] 

r = -40143 + -01721 cos 4( + -00127[ cos St, 
■4019 -0168] -0013 J 







Fi8 


4a. 






?~"^ 


/ 








N, ^"%\ 




. 7 








s s^ 


/•- 






■"•^^ 




" ^ 




/ / 






V 




^ / 


s z 








s 5s' ■■-'•■ 


\9 


tX7 








Sii ^? . 


7l 


7 . 


,^'^ 


■--- 




''■^i$ 


/ 


7 -y 






N 


Z H-\ 


' / 


'_7\ 








\ /\ r 


I 


1 








y V 


\ 


\ 


,-/ 


'■•■. 

\ 




A A 


s \ 


_ \^ 






•■,_ 


/ \l _v 


\ 


\ /N, 






/ 


fe 2\ 7" 


. N, 


^ ••- 




^^^ 




-../ \/ 


^ 


-'^^ 








2i 7\ 


J \ 


/ S 








2 -y \' 


^ -' 


i S 






^ 


^ 7-.- ?; 


. ^''' 


i 




^ 




7 \/ 


. -A 


s 








z ?•• " 


K-^ 


v-^- 






—p 


2 ^z:>; 



the centre of the square being pole. From these we obtain for 
the radius vector parallel to the aides of the square (t - 0) -41980, 

' Od the nodal linei ot » «qnu-e pUt«. Phil. Mag. Angiut, I9T8. 
• Pogg. Ann. Vol. cilvi. p. 819, 1873. 



376 



VIBRATIONS OF PLATES. 



[226 



•41981, '4200, while the calculated result is '4154. The radius 
vector measured along a diagonal is *3856, *d855, *3864, and by 
calculation '3900. 

By crossing the network in the other direction we obtain the 
locus of points for which Wj {xja) — Ui (y/a) is constant, which are 
the curves of constant displacement for that mode in which the 
diagonals are nodal. The pitch of the vibration is (according to 
theory) the same in both cases. 



^ 




Fig. 


48. 






/ 


( 




• 


^S. 


^^ • 


/ 




\\ 




\ 




\ 




/ 




1 


\ 


} 


/ 










\ 



The primitive modes represented by w = m, (a:/a) or w =: t*, (y/a) 

may be combined in like manner. Fig. 43 shews the nodal curve 

for the vibration 

w = ita (a?/a) ± li, (y/a) (5). 

The form of the curve is the same relatively to the other diagonal, 
if the sign of the ambiguity be altered. 



227.] wheatstone's figures. 377 

227. The method of superposition does not depend for its 
application on any particular form of normal function. Whatever 
the form may be, the mode of vibration, which when /tt = 
passes into that just discussed, must have the same period, 
whether the approximately straight nodal lines are parallel to 
a? or to y. If the two synchronous vibrations be superposed, 
the resultant has still the same period, and the general course 
of its nodal system may be traced by means of the considera- 
tion that no point of the plate can be nodal at which the 
primitive vibrations have the same sign. To determine exactly 
the line of compensation, a complete knowledge of the primitive 
normal functions, ^nd not merely of the points at which they 
vanish, would in general be necessary. Doctor Young and the 
brothers Weber appear to have had the idea of superposition as 
capable of giving rise to new varieties of vibration, but it is to Sir 
Charles Wheatstone ^ that we owe the first systematic application 
of it to the explanation of Chladni's figures. The results actually 
obtained by Wheatstone are however only very roughly applicable 
to a plate, in consequence of the form of normal function implicitly 
assumed. In place of Fig. 42 (itself, be it remembered, only an 
approximation) Wheatstone finds for the node of the compound 
vibration the inscribed square shewn in Fig. 44. Fig. 44. 

This form is really applicable, not to a plate vi- 
brating in virtue of rigidity, but to a stretched 
membrane, so supported that every point of the 
circumference is free to move along lines perpen- 
dicular to the plane of the membrane. The 
boundary condition applicable under these circumstances is 

, = ; and it is easy to shew that the normal functions which 

involve only one co-ordinate are 

w =s cos (m7ra?/a), or w == cos (rmry/ a), 

the origin being at a comer of the square. Thus the vibration 

2'n'x . 27ry ._. 

w = coQ — -I- cos- — - (1) 

has its nodes determined by 

TT (a? + y) Trix'-y) ^ 

cos ^ ^cos ^ ^^-Q, 

a a 

1 PHI. Trans. 1888. 




378 



VIBRATIONS OF PLATES. 



[227. 



whence a? + y = ia or |a, ora? — y = + Ja, equations which 
represent the inscribed square. 



If 



w = cos cos — - 

a a 



(2), 



the nodal system is composed of the two diagonals. This result, 
which depends only on the symmetry of the normal functions, is 
strictly applicable to a square plate. 

When m = 3, 

(3), 



37ra? . Ziry 

w = cos h cos — - 

a a 



and the equations of the nodal lines are 



a 5a 



a 



«?-y = ±o» 



shewn in Fig. 45. If the other sign be taken, we 
obtain a similar figure with reference to the other 
diagonal. 




When wi = 4, 



47r/c 47ry 

w = cos h cos 



a 



a 



(4), 



Fig. 46. 



giving the nodal lines 

a Sa 5a 7a .a . 3a .^, .^. 

^■*"^"4' T'T* 4"' *"y=±4' ±-4(F^?*6)- 




With the other sign 



we obtain 



47ra? 47ry 

w = cos cos — - 



a 



a 




« + y = 2'^'T' a?--y=0, ±2 (Fig. 47), 

representing a system composed of the diagonals, 
together with the inscribed square. 

These forms, which are strictly applicable to the membrane, 
resemble the figures obtained by means of sand on a square plate 
more closely than might have been expected. The sequence of 
tones is however quite diflferent. From § 176 we see that, if fi were 
zero, the interval between the form (43) derived from three 
primitive nodes, and (41) or (42) derived from two, would be 



227.] GRAVEST MODE OF SQUARE PLATE, 379 

1*4629 octaves; and the interval between (41) or (42) and (46) or 
(47) would be 2*43o8 octaves. Whatever may be the value of /x the 
forms (41) and (42) should have exactly the same pitch, and the 
same should be true of (46) and (47). With respect to the first- 
mentioned pair this result is not in agreement with Chladni's 
observations, who found a difference of more than a whole tone, 
(42) giving the higher pitch. If however (42) be left out of 
account, the comparison is more satisfactory. According to theorj' 
(m = 0), if (41) gave d, (43) should give g' - , and (46), (47) 
should give (7" + . Chladni found for (43) 5r'jf + , and for (46), 
(47) g'% and g*'t-\- respectively. 

228. The gravest mode of a square plate has yet to be consi- 
dered. The nodes in this case are the two lines drawn through the 
middle points of opposite sides. That there must be such a mode 
will be shewn presently from considerations of symmetry, but 
neither the form of the normal function, nor the pitch, has yet 
been determined, even for the particular case of /a = 0. A rough 
calculation however may be founded on an assumed type of 
vibration. 

If we take the nodal lines for axes, the form w^xy satisfies 
V*w =: 0, as well as the boundary conditions proper for a free edge 
at all points of the perimeter except the actual comers. This is 
in fact the form which the plate would assume if held at rest by 
four forces numerically equal, acting at the comers perpendicu- 
larly to the plane of the plate, those at the ends of one diagonal 
being in one direction, and those at the ends of the other diagonal 
in the opposite direction. From this it follows that w^xycospt 
would be a possible mode of vibration, if the mass of the plate 
were concentrated equally in the four comers. By (3) § 214, we 
see that 

^=3-(rr^r'^'-p* ^i>' 

inasmuch as 

d^wjdx"^ = d^w/dy^ = 0, d^wjdxdy = Qospt 

For the kinetic energy, if p be the volume density, and M the 
additional mass at each comer, 

y = ijp» sin^pt W^'' l^'' 2phahf^dxdy H- \Ma^\ 



-W^^pt[^^l + %M] (2). 



380 VIBRATIONS OF PLATES. [228. 

Hence 

1 _pil+ti)a*/. . -.if' 



1 _p{l+ti)a*/ M\ ,„. 



where if' denotes the mass of the plate without the loads. This 
result tends to become accurate when M is relatively great ; other- 
wise by § 89 it is sensibly less than the truth. But even when 
Jf = 0, the error is probably not very great. In this case we 
should have 

^=._9M'_ (4) 

giving a pitch which is somewhat too high. The gravest mode 
next after this is when the diagonals are nodes, of which the pitch, 
if /Lt = 0, would be given by 

„ ^ qh^ (4'7300y . 

^ "pa' 3 " ^ ^' 

(see § 174). 

We may conclude that if the material of the plate were sucli 
that /A = 0, the interval between the two gravest tones would 
be somewhat greater than that expressed by the ratio 1*318. 
Chladni makes the interval a fifth. 

229. That there must exist modes of vibration in which 
the two shortest diameters are nodes may be 
inferred from such considerations as the following. 
In Fig. (48) suppose that GH is a plate of which 
the edges HO, 00 are supported, and the edges 
OC, CH free. This plate, since it tends to a 
definite position of equilibrium, must be capable 
of vibrating in certain fundamental modes. Fixing 
our attention on one of these, let us conceive a 
distribution of w over the three remaining quadrants, such that in 
any two that adjoin, the values of w are equal and opposite at 
points which are the images of each other in the line of separation. 
If the whole plate vibrate according to the law thus determined, 
no constraint will be required in order to keep the lines OE, FH 
fixed, and therefore the whole plate may be regarded as free. The 
same argument may be used to prove that modes exist in which 
the diagonals are nodes, or in which both the diagonals and the 
diameters just considered are together nodal. 




229.] 



PRINCIPLE OF SYMMETRY. 



381 



The principle of symmetry may also be applied to other forms 
of plate. We might thus infer the possibility of nodal diameters 
in a circle, or of nodal principal axes in an ellipse. When the 

Fig. 49. Fig. 60. Fig. 61. 






boundary is a regular hexagon, it is easy to see that Figs. (49), 
(50), (51) represent possible forms. 

It is interesting to trace the continuity of Chladni's figures, as 
the form of the plate is gradually altered. In the circle, for 
example, when there are two perpendicular nodal diameters, it is a 
matter of indifference as respects the pitch and the type of vibra- 
tion, in what position they be taken. As the circle develops into 
a square by throwing out comers, the position of these diameters 
becomes definite. In the two alternatives the pitch of the vibra- 
tion is different, for the projecting comers have not the same effi- 
ciency in the two cases. The vibration of a square plate shewn in 
Fig. (42) corresponds to that of a circle when there is one circular 
node. The correspondence of the graver modes of a hexagon or 
an ellipse with those of a circle may be traced in like manner. 

230. For plates of uniform material and thickness and of 
invariable shape, the period of the vibration in any fundamental 
mode varies as the square of the linear dimension, provided of 
course that the boundary conditions are the same in all the cases 
compared. When the edges are clamped, we may go further 
and assert that the removal of any external portion is attended 
by a rise of pitch, whether the material and the thickness be 
uniform, or not. 

Let AB be a part of a clamped edge (it is of no consequence 
whether the remainder of the boundary be clamped, or not), and 



382 VIBRATIONS OF PLATES. [230. 

let the piece ACBD be removed, the new edge ADB being also 
clamped. The pitch of any fundamental vibration is sharper 
than before the change. This is evident, since the altered 
vibrations might be obtained from the original system by the 
introduction of a constraint clamping the edge ADB, The effect 
of the constraint is to raise the pitch of every component, and 
the portion ACBD being plane and at rest throughout the motion, 
may be removed. In order to follow the sequence of changes 
with greater security from error, it is best to suppose the line 
of clamping to advance by stages between the two positions 
ACB, ADB, For example, the pitch of a uniform clamped plate 
in the form of a regular hexagon is lower than for the inscribed 
circle and higher than for the circumscribed circle. 

When a plate is free, it is not true that an addition to 
the edge always increases the period. In proof of this it may be 
sufficient to notice a particular case. 

AB is a narrow thin plate, itself without inertia but carrying, 
loads B,t A, By C, It is clear that the addition to the breadth 

Fig. 53. 







^ 


•A. 


•C 


B* 


1 







indicated by the dotted line would augment the stiffness of the 
bar, and therefore lessen the period of vibration. The same 
consideration shews that for a uniform free plate of given area 
there is no lower limit of pitch ; for by a sufficient elongation 
the period of the gravest component may be made to exceed 
any assignable quantity. When the edges are clamped, the 
form of gravest pitch is doubtless the circle. 

If all the dimensions of a plate, including the thickness, be 
altered in the same proportion, the period is proportional to the 
linear dimension, as in every case of a solid body vibrating in 
virtue of its own elasticity. 

The period also varies inversely as the square root of Young's 
modulus, if fi be constant, and directly as the square root of the 
mass of unit of volume of the substance. 



231.] CYLINDER OF RING. 383 

231. Experimenting with square plates of thin wood whose 
grain ran parallel to one pair of sides, Wheatstone^ found that 
the pitch of the vibrations was different according as the ap- 
proximately straight nodal lines were parallel or perpendicular 
to the fibre of the wood. This effect depends on a variation 
in the flexural rigidity in the two directions. The two sets of 
vibrations having different periods cannot be combined in the 
usual manner, and consequently it is not possible to make such 
a plate of wood vibrate with nodal diagonals. The inequality 
of periods may however be obviated by altering the ratio of the 
sides, and then the ordinary mode of superposition giving nodal 
diagonals is again possible. This was verified by Wheatstone. 

A further application of the principle of superposition is due 
to Konig*. In order that two modes of vibration may combine, 
it is only necessary that the periods agree. Now it is evident 
that the sides of a rectangular plate may be taken in such a 
ratio, that (for instance) the vibration with two nodes parallel 
to one pair of sides may agree in pitch with the vibration having 
three nodes parallel to the other pair of sides. In such a case 
new nodal figures arise by composition of the two primary modes 
of vibration. 

232. When the plate whose vibrations are to be considered 
is naturally curved, the difficulties of the question are generally 
much increased. But there is one case in which the complication 
due to curvature is more than compensated by the absence of a 
free edge ; and this case happens to be of considerable interest, as 
being the best representative of a bell which admits of simple 
analytical treatment. 

A long cylindrical shell of circular section and uniform thick- 
ness is evidently capable of vibrations of a flexural character 
in which the axis remains at rest and the surface cylindrical, 
while the motion of every part is perpendicular to the generating 
lines. The problem may thus be treated as one of two dimensions 
only, and depends upon the consideration of the potential and 
kinetic energies of the various deformations of which the section 
is capable. The same analysis also applies to the corresponding 
vibrations of a ring, formed by the revolution of a small closed 
area about an external axis (§ 192 a). 

1 Phil Trans, 1833. 

3 Pogg. ^7171. 1884, czxii. p. 288. 



384 VIBRATIONS OF PLATES. [232. 

The cylinder, or ring, is susceptible of two classes of vibrations 
depending respectively on extensibility and flexnral rigidity, and 
analogous to the longitudinal and lateral vibrations of straight 
bars. When, however, the cylinder is thin, the forces resisting 
bending become small in comparison with those by which ex- 
tension is opposed; and, as in the case of straight bars, the 
vibrations depending on bending are graver and more important 
than those which have their origin in longitudinal rigidity. 
In the limiting case of an infinitely thin shell (or ring), the 
flexural vibrations become independent of any extension of the 
circumference as a whole, and may be calculated on the sup- 
position that each part of the circumference retains its natural 
length throughout the motion. 

But although the vibrations about to be considered are 
analogous to the transverse vibrations of straight bars in respect 
of depending on the resistance to flexure, we must not fall into 
the common mistake of supposing that they are exclusively 
normal. It is indeed easy to see that a motion of a cylinder or 
ring in which each particle is displaced in the direction of the 
radius would be incompatible with the condition of no extension. 
In order to satisfy this condition it is necessary to ascribe to 
each part of the circumference a tangential as well as a normal 
motion, whose relative magnitudes must satisfy a certain differ- 
ential equation. Our first step will be the investigation of this 
equation. 

233. The original radius of the circle being a, let the equi- 
librium position of any element of the circumference be defined 
by the vectorial angle 0, During the motion let the polar co- 
ordinates of the element become 

r = a + Sr, <f>='0 + 80. 

If ds represent the arc of the deformed curve corresponding to add, 
we have 

(dsy = (ad0y = (dSry + r^(d0 + dS0y ; 

whence we find, by neglecting the squares of the small quantities 
8r, 80, 

Br dS0 ^ 

T^W=' <!)• 

as the required relation. 



233.] POTENTIAL AND KINETIC ENERGIES 385 

In whatever manner the original circle may be deformed at 
time t, Sr may be expanded by Fourier's theorem in the series 

Sr = a {^icos ^ + 5isin ^ + iljCos 2^ + 5asin 2^+ ... 

'\'AtCos80-\-Btsm80+...} (2), 

and the corresponding tangential displacement required by the 
condition of no extension will be 

A B 

S0=^-'A,sm0 + B,cos0+... ''Siu80+ — coQ80- (3), 

8 8 

the constant that might be added to 80 being omitted. 

If <rad0 denote the mass of the element ad0, the kinetic 
energy T of the whole motion will be 






+ (l + i) (i.' + M + ...| (4), 



the products of the co-ordinates Ag, B, disappearing in the 
integration. 

We have now to calculate the form of the potential energy V. 
Let p be the radius of curvature of any element ds ; then for the 
corresponding element of V we may take ^Bd8 {S (1/p)}*, where 
S is a constant depending on the material and on the thickness. 
Thus 



Now 
and 



= i5a r7s-Yrfd (5). 



llp = u + d^u/d<f>^, 
M = - = - {1 — ill cos <^ — -Bi sin <^ — . . . }, 



for in the small terms the distinction between <^ and may be 
neglected. 

Hence 

8 - = - 2 1 («» - 1 ) (^, cos «^ + 5, sin «<^) }, 
p a 

R. 25 



386 VIBBATIONS OF PLATES. [233. 

and 

F= ^r {2 («» - 1) (A, COS sd + B, sin 80)Yde 

= TT ^ 2 («« - !)• (4.' + 5.») (6). 

in which the summation extends to all positive integral values 
of 8. 

The term for which « = 1 contributes nothing to the potential 
energy, as it corresponds to a displacement of the circle as a whole, 
without deformation. 

We see that when the configuration of the system is defined as 
above by the co-ordinates Ai, Bi, &c., the expressions for T and V 
involve only squares; in other words, these are the normal co- 
ordinates, whose independent harmonic variation expresses the 
vibration of the system. 

If we consider only the terms involving cos 80, sin 80, we have 
by taking the origin of suitably, 

Sr = aA,coa80, 80^ ^sin«^ (7), 

8 

while the equation defining the dependence of A, upon the 
time is 

<ra»(l + A)i. + ^(s._l)M. = ....(8), 

from which we conclude that, if A, varies as cos (pt — e), 

This result was given by Hoppe for a ring in a memoir pub- 
lished in Crelle, Bd. 63, 1871. His method, though more complete 
than the preceding, is less simple, in consequence of his not re- 
cognising explicitly that the motion contemplated corresponds to 
complete inextensibility of the circumference. 

[In the application of (9) to a ring we have, § 192 a, 
where q is Young's modulus, p the volume density, and c the 



233.] EQUATION FOR THE FREQUENCY. 887 

radius of the circular section. For the cylindrical shell, (18) 

§ 235 fir, 

B 4mnh* , . 

^"3(m + w)p ^ ^' 

2h denoting the thickness, and m, n the elastic constants in 
Thomson and Tait's notation.] 

According to Chladni the frequencies of the tones of a ring 

are as 

3* : 5» : 7« : 9^ ....;.... 

If we refer each tone to the gravest of the series, we find for 
the ratios characteristic of the intervals 

2-778, 5-445, 9, 13-44, &c. 

The corresponding numbers obtained from the above theoretical 
formula (9), by making 8 successively equal to 2, 3, 4, &c., are 

2-828, 5-423, 8-771, 1287, &c., 

agreeing pretty nearly with those found experimentally. 

[Observations upon the tones of thin metallic cylinders, open 
at one end, have been made by Fenkner \ Since the pitch proved 
to be very nearly independent of the height of the cylinders, the 
vibrations may be regarded as approximately two-dimensional. 
In accordance with (9), (11), Fenkner found the frequency propor- 
tional to the thickness directly, and to the square of the radius 
inversely. As regards the sequence of tones from a given 
cylinder ^ the numbers, referred to the gravest (« = 2) as unity, 
were 267, o'OO, 8*00, 12*00, &c. The agreement with (9) would 
be improved if these numbers were raised by about -^ part, 
equivalent to an alteration in the pitch of the gravest tone. 

The influence of rotation of the shell about its axis has been 
examined by Bryan'. It appears that the nodes are carried 
round, but with an angular velocity less than that of the rotation. 
If the latter be denoted by g>, the nodal angular velocity is 

«»-l , 

S* + l -■ 

1 Wied. Ann. vol. 8, p. 186, 1879. 

s Melde, Akustik, Leipzig, 1883, p. 223. 

8 Proc. Camb, Phil. Soc. vol. vn. p. 101, 1890. 

26—2 



388 VIBBATIONS OF PLATES. [234. 

234. When « = 1, the frequency is zero, as might have been 
anticipated. The principal mode of vibration corresponds to « = 2, 
and has four nodes, distant from each other by 90*. These so- 
called nodes are not, however, places of absolute rest, for the 
tangential motion is there a maximum. In fact the tangential 
vibration at these points is half the maximum normal motion. 
In general for the 8^ term the maximum tangential motion is 
(1/^) of the maximum normal motion, and occurs at the nodes of 
the latter. 

When a bell-shaped body is sounded by a blow, the point of 
application of the blow is a place of maximum normal motion 
of the resulting vibrations, and the same is true when the 
vibrations are excited by a violin-bow, as generally in lecture- 
room experiments. Bells of glass, such as finger-glasses, are 
however more easily thrown into regular vibration by friction with 
the wetted finger carried round the circumference. The pitch of 
the resulting sound is the same as of that elicited by a tap with 
the soft pai-t of the finger; but inasmuch as the tangential motion 
of a vibrating bell has been very generally ignored, the production 
of sound in this manner has been felt as a difficulty. It is now 
scarcely necessary to point out that the eflfect of the friction is in 
the first instance to excite tangential motion, and that the point 
of application of the friction is the place where the tangential 
motion is greatest, and therefore where the normal motion 
vanishes. 

236. The existence of tangential vibration in the case of a bell 
was verified in the following manner. A so-called air-pump re- 
ceiver was securely fastened to a table, open end uppermost, and set 
into vibration with the moistened finger. A small chip in the rim, 
reflecting the light of a candle, gave a bright spot whose motion 
could be observed with a Coddington lens suitably fixed. As the 
finger was carried round, the line of vibration was seen to re- 
volve with an angular velocity double that of the finger; and 
the amount of excursion (indicated by the length of the line of 
light), though variable, was finite in every position. There was, 
however, some difficulty in observing the correspondence between 
the momentary direction of vibration and the situation of the point 
of excitement. To effect this satisfactorily it was found necessary 
to apply the friction in the neighbourhood of one point. It then 
became evident that the spot moved tangentially when the bell was 



235.] TANGENTIAL MOTION. 389 

excited at points distant therefrom 0, 90, 180, or 270 degrees ; and 
normally when the friction was applied at the intermediate points 
corresponding to 45, 135, 226 and 315 degrees. Care is sometimes 
required in order to make the bell vibrate in its gravest mode 
without sensible admixture of overtonea 

If there be a small load at any point of the circumference, 
a slight augmentation of period ensues, which is different accord- 
ing as the loaded point coincides with a node of the normal or 
of the tangential motion, being greater in the latter case than 
in the former. The sound produced depends therefore on the 
place of excitation; in general both tones are heard, and by 
interference give rise to beatSy whose frequency is equal to the 
difference between the frequencies of the two tones. This phe- 
nomenon may often be observed in the case of large bells. 

236 a. In determining the number of nodal meridians (2«) 
corresponding to any particular tone of a bell, advantage may be 
taken of beats, whether due to accidental irregularities or intro- 
duced for the purpose by special loading (compare §§ 208, 209). By 
tapping cautiously round a circle of latitude the places may be in- 
vestigated where the beats disappear, owing to the absence of one 
or other of the component tones. But here a decision must not 
be made too hastily. The inaudibility of the beats may be favoured 
by an unsuitable position of the ear or of the mouth of the re- 
sonator used in connection with the ear. By travelling round, 
a situation is soon found where the observation can be made to 
the best advantage. In the neighbourhood of the place where the 
blow is being tried there is a loop of the vibration which is most 
excited and a (coincident) node of the vibration which is least 
excited. When the ear is opposite to a node of the first vibration, 
and therefore to a loop of the second, the original inequality is 
redressed, and distinct beats may be heard even though the 
deviation of the blow from a nodal point may be very small. The 
accurate determination in this way of two consecutive places where 
no beats are generated is all that is absolutely necessary for the 
purpose in view. The ratio of the entire circumference of the 
circle of latitude to the arc between the points in question is in 
fact 4«. Thus, if the arc between consecutive points proved to 
be 45°, we should infer that we were dealing with the case of « = 2, 
in which the deformation is elliptical. As a greater security 
against error, it is advisable in practice to determine a larger 



390 VIBRATIONS OF PLATES. [235 a. 

number of points where no beats occur. Unless the deviation 
from symmetry be considerable, these points should be uniformly 
distributed along the circle of latitude ^ 

In the above process for determining nodes we are supposed to 
hear distinctly the tone corresponding to the vibration under 
investigation. For this purpose the beats are of assistance in 
directing the attention; but in dealing with the more difficult 
subjects, such as church bells, it is advisable to have recourse to 
resonators. A set of v. Helmholtz's pattern, as manufactured by 
Konig, are very convenient. The one next higher in pitch to 
the tone under examination is chosen and tuned by advancing the 
finger across the aperture. Without the security afforded by 
resonators, the determination of the octave is very uncertain. 

The only class of bells, for which an approximate theory can 
be given, are those with thin walls, §§ 233, 235 c. Of such the 
following glass bells may be regarded as examples : — 

I. c', e"t?, &'% 

II. a, c'% V\ 
in. f% 6". 

The value of s for the gravest tone was 2, for the second 3, 
and for the third tone 4. 

Similar observations have been made upon a so-called hemi- 
spherical bell, of nearly uniform thickness, and weighing about 3 
cwt. Four tones could be plainly heard, 

e^ f% e", h", 

the pitch being taken from a harmonium. The gravest tone has a 
long duration. When the bell is struck by a hard body, the 
higher tones are at first predominant, but after a time they die 
away, and leave e^ in possession of the field. If the striking body 
be soft, the original preponderance of the higher elements is less 
marked. 

By the method described there was no difficulty in shewing 
that the four tones correspond respectively to « = 2, 3, 4, 5. Thus 
for the gravest tone the vibration is elliptical with 4 nodal meri- 
dians, for the next tone there are 6 nodal meridians, and so on. 

^ The bells, or gODgs, as they are sometimes caUed, of striking clocks often give 
disagreeable beats. A remedy may be found in a suitable rotation of the beU round 
its axis. 



235 a.] BELLS. 391 

Tapping along a meridian shewed that the sounds became less 
clear as the edge was departed from, and this in a continuous 
manner ¥dth no suggestion of a nodal circle of latitude. A question 
to which we shall recur in connection with church bells here 
suggests itself. Which of the various coexisting tones characterizes 
the pitch of the bell as a whole ? It would appear to be the third 
in order, for the founders gave the pitch as E natural. 

In church bells there is great concentration of metal at the 
*^ sound-bow " where the clapper strikes, indeed to such an extent 
that we can hardly expect much correspondence with what occurs 
in the case of thin uniform bells. But the method already 
described suffices to determine the number of nodal meridians for 
all the more important tones. From a bell of 6 cwt. by Mears 
and Stainbank 6 tones could be obtained, viz. : 

«'. c", /"+, 6"t>. d'", /'". 
(4) (4) (6) (6) (8) 

The pitch of this bell as given by the makers is df\ so that it 
is the fifth in the above series of tones which characterizes the 
bell. The number of nodal meridians in the various components 
is indicated within the parentheses. Thus in the case of the tone 
e' there are 4 nodal meridians. A similar method of examination 
along a meridian shewed that there was no nodal circle of latitude. 
At the same time differences of intensity were observed. This 
tone is most fully developed when the blow is delivered about 
midway between the crown and the rim of the bell. 

The next tone is c". Observation shewed that for this vibra- 
tion also there are four, and but four, nodal meridians. But now 
there is a well-defined nodal circle of latitude, situated about a 
quarter of the way up from the rim towards the crown. As heard 
with a resonator, this tone disappears when the blow is accurately 
delivered at some point of this circle, but revives with a very small 
displacement on either side. The nodal circle and the four meri- 
dians divide the surface into segments, over each of which the 
normal motion is of one sign. 

To the tone /" correspond 6 nodal meridians. There is no 
well-defined nodal circle. The sound is indeed very fiednt when 
the tap is much displaced from the sound-bow; it was thought 
to fall to a minimum when a position about half-way up was 
reached. 



392 VIBRATIONS OF PLATES. [235 a. 

The three graver tones are heard loudly fix)m the sound-bow. 
But the next in order, V]>, is there scarcely audible, unless the 
blow is delivered to the rim itself in a tangential direction. The 
maximum effect occurs about half-way up. Tapping round the 
circle revealed 6 nodal meridiana 

• ^^ 

The fifth tone, d'", is heard loudly from the sound-bow, but 

soon falls off when the locality of the blow is varied, and in the 

upper three-fourths of the bell it is very faint. No distinct circular 

node could be detected. Tapping round the circumference shewed 

that there were 8 nodal meridians. 

The highest tone recorded, /'", was not easy of observation, 
and the mode of vibration could not be fixed satisfactorily. 

Similar results have been obtained fi:om a bell of 4 cwt., cast 
by Taylor of Loughborough for Ampton church. The nominal 
pitch (without regard to octave) was d, and the following were the 
tones observed : — 

e1^-2, d"~6, /' + 4, 6"b— 6", d'", g"\ 
(4) (4) (6) (6) (8) 

In the specification of pitch the numerals following the note 
indicate by how much the frequency for the bell differed ftx)m 
that of the harmonium employed as a standard. Thus the gravest 
tone e]> gave 2 beats per second, and was flat. When the number 
exceeds 3, it is the result of somewhat rough estimation, and 
cannot be trusted to be quite accurate. Moreover, as has been 
explained, there are in strictness two frequencies under each 
head, and these often differ sensibly. In the case of the 4th tone, 
6"b — 6" means that, as nearly as could be judged, the pitch of the 
bell was midway between the two specified notes of the 
harmonium. 

Observations in the laboratory upon the above-mentioned bells 
having settled the modes of vibration corresponding to the five 
gravest tones, other bells of the church pattern could be suflSciently 
investigated by simple determinations of pitch. The results are 
collected in the following tabled and include, besides those already 
given, observations upon a Belgian bell, the property of Mr 
Haweis, and upon the five bells of the Terling peal. As regards 

1 On Bells, Phil, Mag., vol. 29, p. 1, 1S90. 



235 a.] 



CHURCH BELLS. 



393 



the nominal pitch of the latter bells, several observers concurred 
in fixing the notes of the peal as 

ft 5*» ^% &> cj{, 

no attention being paid to the question of the octave. 



Mean, 


Amptom, 
1888. 


Balgum 

TIaII 


Terling (5),iTerling (4), Terling (8), Terling (2), Terling (1), 
Osbom, 1 Hears, Graye, Oardner, Warner, 


xooo. 


001I« 


1788. 1810. 


1628. 


1728. 


1868. 


Actual Pitch by Uarmonium. 


e' 


e't?-2 


«r-4 i7-3 


a+3 


ajf+3 


(f-e 


<f + 2 


c" 


cf'-6 


c"»-«r' /-4 


5^»-4 


a'+6 


a'tf-6 


6' + 2 


/" + 


/'+4 


/"+! 1 o'+6 


fc'+e 


^'»+4 


<f'+8 


e" 


6"t? 


h"^-h" 


a"-6 «r'-3 


^'»-«" 


«"+6 


<7"»+ao) 


5^'»+4 


d" 


d" 




/'»-2 


5^'»-6 


«"» 


6"+2 


C"tf+3 


r 


f' 














Pitch referred to fifth tone as c. 


d 


cJP-2 




ctf-3 


cJP+3 


c+3 


et?-6 


ctf+2 


6t? 


c-6 




ctf-4 


c-4 


6t?+6 


6-6 


6t?+2 


et? + 


. ct?+4 




et?+6 


«t?+6 


ct? + 4 


el? +8 


et? 


at? 


at? -a 




at?-3 


9-SfS 


/tt+6 


a+8 


^+4 


c 


c 


<r-2 


c-6 


c 


c-f2 


c+3 



Examination of the table reveals the remarkable fact that 
in every case of the English bells it is the 5th tone in order 
which agrees with the nominal pitch, and that, with the exception 
of Terling (4), no other tone shews such agreements Moreover, 
as appeared most clearly in the case of the bell cast by Mears and 
Stainbank, the nominal pitch, as given by the makers, is an octave 
below the only corresponding tone. 

The highly composite, and often discordant, character of the 
sounds of bells tends to explain the discrepancies sometimes 
manifested in estimations of pitch. Mr Simpson, who has devoted 
much attention to the subject, has put forward strong arguments 
for the opinion that the Belgian makers determine the pitch of 
their bells by the tone 2nd in order in the above series, so that 
for instance the pitch of Terling (3) would be a and not at. In 
subordination to this tone they pay attention also to the next 
(the 3rd in order), classifying their bells according to the character 

^ In this oomparison the gravest tone is disregarded. 



394 VIBRATIONS OF PLATES. [235 a. 

of the third, whether major or minor, so compounded. Thus 
in TerUng (3) the interval, a' to c" , is a mxijor third. The com- 
parative neglect with which the Belgians treat the 5th tone, 
regarded almost exclusively by English makers, may perhaps be 
explained by a less prominent development of this tone in Belgian 
bells, and by a difference in treatment. When a bell is sounded 
alone, or with other bells in a comparatively slow succession, 
attention is likely to concentrate itself upon the graver and more 
persistent elements of the sound rather than upon the acuter 
and more evanescent elements, while the contrary may be 
expected to occur when bells follow one another rapidly in a peal. 

In any case the false octaves with which the Table abounds 
are simple facts of observation, and we may well believe that their 
correction would improve the general effect. Especially should 
the octave between the 2nd tone and the 5t^ tone be made true. 
Probably the lower octave of the gravest, or hunirnote, as it is 
called by English founders, is of less importance. The same may 
be said of the fifth, given by the 4th tone of the series, which 
is much less prominent. The variations recorded in the Table 
would seem to shew that no insuperable obstacle stands in the 
way of obtaining accurate harmonic relations among the various 
tones. 

No adequate explanation has been given of the form adopted 
for church belLs. It appears both from experiment and from the 
theory of thin shells that this form is especially stiff, as regards the 
principal mode of deformation {s = 2), to forces applied normally 
and near the rim. Possibly the advantage of this form lies in its 
rendering less prominent the gravest component of the sound, 
or the hum-note. 



CHAPTER Xa. 

CUBVED PLATES OB SHELLS. 

236 6. In the last chapter (§§ 232, 233) we have considered 
the comparatively simple problem of the vibration in two dimen- 
sions of a cylindrical shell, so far at least as relates to vibrations 
of a flexural character. The shell is supposed to be thin, to be 
composed of isotropic material, and to be bounded by infinite 
coaxal cylindrical surfaces. It is proposed in the present chapter 
to treat the problem of the cylindrical shell more generally, and 
further to give the theory of the flexural vibrations of spherical 
shells. 

In considering the deformation of a thin shell the most 
important question which presents itself is whether the middle 
surface, viz. the surface which lies midway between the boundaries, 
does, or does not, undergo extension. In the former case the 
deformation may be called extensional, and its potential energy is 
proportional to the thickness of the shell, which will be denoted 
by 2A. Since the inertia of the shell, and therefore the kinetic 
energy of a given motion, is also proportional to A, the frequencies 
of vibration are in this case independent of h, § 44. On the 
other hand, when no line traced upon the middle surface under- 
goes extension, the potential energy of a deformation is of a 
higher order in the small quantity h. If the shell be conceived 
to be divided into laminae, the extension in any lamina is pro- 
portional to its distance from the middle surface, and the con- 
tribution to the potential energy is proportional to the square 
of that distance. When the integration over the thickness 
is carried out, the whole potential energy is found to be propor- 
tional to h\ Vibrations of this kind may be called inextensional, 



396 CURVED PLATES OR SHELLS. [235 6. 

or flexural, and (§ 44) their frequencies are proportional to h, so 
that the sounds become graver without limit as the thickness is 
reduced. 

Vibrations of the one class may thus be considered to depend 
upon the term of order h, and vibrations of the other class upon 
the term of order A*, in the expression for the potential energy. 
In general both terms occur ; and it is only in the limit that the 
separation into two classes becomes absolute. This is a question 
which has sometimes presented difficulty. That in the case of 
extensional vibrations the term in h* should be negligible in 
comparison with the term in h seems reasonable enough. But 
is it permissible in dealing with the other class of vibrations to 
omit the term in h while retaining the term in A' ? 

The question may be illustrated by considei-ation of a statical 
problem. It is a general mechanical principle (§ 74) that, if given 
displacements (not sufficient by themselves to determine the 
I configuration) be produced in a system originally in equilibrium 
by forces of corresponding types, the resulting deformation is 
determined by the condition that the potential energy shall be 
as small as possible. Apply this principle to the case of an elastic 
shell, the given displacements being such as not of themselves to 
involve a stretching of the middle surface. The resulting defor- 
mation will, in general, include both stretching and bending, and 
any expression for the energy will be of the form 

Ah (extension)* + Bh^ (bending)' (1). 

This energy is to be as small as possible. Hence, when the 
thickness is diminished without limit, the actual displacement 
will be one of pure bending, if such there be, consistent with 
the given conditions. 

At first sight it may well appear strange that of the two terms 
the one proportional to the cube of the thickness is to be retained, 
while that proportional to the first power may be neglected. The 
fact, however, is that the large potential energy that would 
accompany any stretching of the middle surface is the very reason 
why such stretching does not occur. The comparative largeness 
of the coefficient (proportional to h) is more than neutralized by 
the smallness of the stretching itself, to the square of which the 
energy is proportional. 



235 6.] FLEXUBAL VIBRATIONS. 397 

An example may be taken from the case of a rod, clamped at 
one end A, and deflected by a lateral force ; it is required to trace 
the effect of constantly increasing stiffness of the part included 
between A and a neighbouring point B, In the limit we may 
regard the rod as clamped at B, and neglect the energy of the 
part A By in spite of, or rather in consequence of, its infinite 
stiflhess. 

It would thus be a mistake to regard the omission of the term 
in A as especially mysterious. In any case of a constraint which 
is supposed to be giudually introduced (§ 92 a), the vibrations ^ 
tend to arrange themselves into two classes, in one of which the 
constraint is observed, while in the other, in which the constraint 
is violated, the frequencies increase without limit. The analogy 
with the shell of gradually diminishing thickness is complete if « 
we suppose that at the same time the elastic constants are in- 
creased in such a manner that the resistance to bending remains 
unchanged. The resistance to extension then becomes infinite, 
and in the limit one class of vibrations is purely inextensional, or ' 
flexural. 

In the investigation which we are about to give of the ' 
vibrations of a cylindrical shell, the extensional and the in- 
extensional classes will be considered separately. It would 
apparently be more direct to establish in the first instance a 
general expression for the potential energy complete as &r as 
the term in A', from which the whole theory might be deduced. 
Such an expression would involve the extensions and the curva- 
tures of the middle surface. It appears, however, that this method 
is difficult of application, inasmuch as the potential energy (correct 
to h?) does not depend only upon the above-mentioned quantities, 
but also upon the manner of application of the normal forces, 
which are in general implied in the existence of middle surface 
extensions*. 

236 c. The first question to be considered is the expression of 
the conditions that the middle surface remain unextended, or if 
these conditions be violated, to find the values of the extensions in 
terms of the displacements of the various points of the sui-face. 

^ On the Uniform Deformation in Two Dimensioips of a Cylindrical Shell, with 
Application to the General Theory of Deformation of Thin Sheila. Proe. Math, 
Soc, vol. zz. p. 872, 18S9. 



398 CUBVED PLATES OR SHELLS. [235 C. 

We ¥dll suppose in the first instance merely that the surface is of 
revolution, and that a point is determined by cylindrical co-ordi- 
nates z, r, <l>. After deformation the co-ordinates of the above 
point become z + Bz, r + Br, ^ + S^ respectively. If ds denote 
an element of arc traced upon the surface, 

(ds + dSsY = (dz + dSzy + (r + &-)« (d<f> + dS<l>y + (dr + dSr)\ 

so that 

dsd8&=-dzdSz + r^if>dS<l> + rSr (d<l>y + dr dBr (1). 

In this we regard z and ^ as independent variables, so that, for 
example, 

,c, dBz J dhz J , 

diT dj* 

while dr = ^dz + ^ d^, 

in which by hypothesis dr/d<j> = 0. Accordingly 
ds {dBYidz'*' dz dz \ ^ (day \ d<f> ^ j 



(2). 



dzd<l> (dSz dS(f) dr dBr 
(ds)* \d^ dz dzd<f>) 

in which dSs/ds represents the extension of the element ds. If 
there be no extension of any ai-c traced upon the surface, (2) must 
vanish independently of any relations between dz and d<^. Hence 

dz dz dz 



(5)- 



d8z dS<l> dr dSr _ ^ 

d<f> dz dz d<f> 

From these, by elimination of Br, 

dBz dr d f dB4>\ _ ^ 

dz dzdz\ d<i> ) ' 

dSz dS<f> drd^B<f,_ 

d4> dz dz d^" ' 

and again, by elimination of hz, 

dz\ dz) dz' d<l>' ^^- 



235 c.] CONDITIONS OP INEXTENSION. 399 

If the distribution of thickness and the form of the boundary 
or boundaries be symmetrical with respect to the axis, the normal 
functions of the system are to be found by assuming B<f> to be 
proportional to cos 8(f>, or sin 8<f>, The equation for 8<t> may then 
be put into the form 



"U'-w)*^"^^" ">■ 



It will be seen that the conditions of inextension go a long way 
towards determining the form of the normal functions. 

The simplest application is to the case of a cylinder for which 
r is constant, equal say to a. Thus (3), (4), (5), (7) become simply 

dBz ^ ^ . d8<b ^ dBz „d8<b ^ ,^. 
,-=0, Sr + a-v^ = 0, :rr + a'-5T^0 (8), 



dz ' d<f> * d<f> dz 

(9). 



^-0 
dz' ' 



By (9), if S<f> oc C08 8<t>, we may take 

aS<f> = (Aga -h Bgz) cos 8<f> (10), 

and then, by (8), 8r = 8 (Aga + B^z) sin 8^ (11), 

8z =:- sr^ Bga sin 8<f> (12). 

Corresponding terms, with fresh arbitrary constcmts, obtained by 
>vriting 8<f} 4- Jtt for 8<f>, may of course be added. If jB, = 0, the 
displacement is in two dimensions only (§ 233). 

If an inextensible disc be attached to the cylinder at 2r = 0, so 
as to form a kind of cup, the displacements 8r and 8^ must vanish 
for that value of z, exception being made of the case 8 = 1. Hence 
Ag = 0, and 

a8<f> = BgZCos8^, 8r = 8 BgZ sin 8^, 8-? = — «"*^£,asin«^...(13). 

Again, in the case of a cone, for which r = tan y . z, the equa- 
tions (3), (4), (5), (7) become 



(14), 



dSz , ^ d8r ^ . d8<f> , ^ ^\ 
Tz'^^^'^^Tz^^' ;.tan7^ + 8r = 0^ 

^ + ^Han^7#^ + tan7^=0 
d<f) dz ' d<f> J 



400 CURVED PLATES OR SHELLS. [235 O 

If we take, as usual, B^ oc cos 8^, we get as the solution of (15) 

8<f> ^ (A, + Bt2r^) cos 8<f> (16), 

and corresponding thereto 

Sr = 8 tSLU y (AgZ -{• Bg) sin 8<f) (17), 

8z = tan» 7 [^-^ jB, - « (A,z + 5,)] sin «i^ (18). 

If the cone be complete up to the vertex at £^ » 0, £« = 0, so that 

S<f> = As cos 8(f) (19), 

Sr = 8Agr^n8<f> (20), 

8-8^ = — «il,tan7rsin«^ (21). 

For the cone and the cylinder, the second term in the general 
equation (7) vanishes. We shall obtain a more extensive class of 
soluble cases by supposing that the surface is such that 

cPr 
r* -j-^ = constant (22), 

an equation which is satisfied by surfaces of the second degree in 
general. If 

a' + P = l (23). 

we shall find r*-=-^=: — - (24); 

and thus (7) takes the form 

d^-^«* = (25), 

if B<f) oc cos 8<f>, and a is defined by 

a=Jr-2d2r (26), 

or in the present case 

« = 26-'^°«a-. (27). 

The solution of (25) is 

H^^y-^^Th-* ^^>- 

The corresponding values of Sr and Bz are to be obtained from (4) 
and (5). 



235 c] NORMAL MODES. 401 

If the surface be complete through the vertex ^ = a, the term 
multiplied by B must disappear. Thus, omitting the constant 
multiplier, we may take 

S* = (^)''co8«^ (29); 

whence, by (4), (5), 

or =» — ;— — ~— T sm «<^ (30). 

8z = (8z + a)^^^^^^sm8il> (31). 

a\a + z)^ ^ ^ ^ 

If we measure z' from the vertex, z'^a — z, and we may write 
S<f>=(^JooBS(f> (32), 

Sr=:8r(^X sin 8<l> (33), 



Bz 



= -S/ = lV(« + l)a-5/U^ysin«<^ (34). 



For the pambola, a and b are infinite, while b^/a = 2a, and 
r^ = 4sa'z\ Thus we may take^ 

S^ = r*cos«<^, Sr = ^+^ sin 5<^, 8^ = — 2(« + l)aVsin«^...(35). 

We will now take into consideration the important case of the 
sphere, for which in (23) b = a. Denoting by the angle between 
the radius vector and the axis, we have z = a cos 0,r = a sin 0, and 
thus from (29), (30), (31) 

8<^ = cos 5<^ tan* i^ (36), 

Sr/a = 8 sin 8<t> sin tan* ^0 (37), 

82r/a = (1 + « cos ^) sin 50 tan* i^ (38). 

The other terms of the complete solution, corresponding to 
(28), are to be obtained by changing the sign of 8, 

In the above equations the displacements are resolved parallel 
and perpendicular to the axis ^ = 0. It would usually be more 
convenient to resolve along the normal and the meridian. If the 
components in these directions be denoted by w and aB0, we have 

1^ = Sr sin ^ -f Bz cos 0, aS0 = Br cos — 8z&m0; 

1 On the Infinitesimal Bending of Sorfaoes of Revolution. Proc. Math, Soc., 
vol. xui. p. 4, 1881. 

R. 26 



402 CURVED PLATES OR SHELLS. [235 C. 

SO that altogether 

S<f>^co8 8<t>[AstsLn*^d-\'BsCoV^e] (39), 

B0 = - sin 8<f} sine [A, tB.n'l^d'-Bs cot' ^6] (40), 

iv/a = sin s^ [As (s + cos d) tan* 1^0 + jB, (« — cos 0) cot* 1^0] . . .(41). 

To the above may be added terms derived by writing ^^ + ^ 
for 8<f>, and changing the arbitrary constants. 

235 d. We now proceed to apply the equations of § 235 c to 

the principal extensions of a cylindrical surface, with a view to the 

formation of the expression for the potential energy. The axial 

and circumferential extensions will be denoted respectively by e^, 

62, and the shear by w. The first of these is given by (2) § 235 c, 

if we suppose that d<f> = 0, dz/ds = 1. Since in the case of a 

cylinder dr/dz = 0, we find 

dBz 

'^^dz ^^>- 

In like manner 

e.= ^%^if (2). 

The value of the shear may be arrived at by considering the 
diflFerence of extensions for the two diagonals of an infinitesimal 
square whose sides are dz and ad^. It is 

a d<f> dz ^' 

The next part of the problem, viz. the expression of the potential 
energy by means of e,, e.^, w, appertains to the general theory of 
elasticity, and can only be treated here in a cursory manner. But 
it may be convenient to give the leading steps of the investigation, 
referring for further explanations to the treatises of Thomson and 
Tait and of Love. In the notation of the former {Natural 
Philosophy, § 694) the general equations in three dimensions are 

na = /S, ?i6 = r, nc^U (4), 

.lf/=Q-cr(i? + P)l (5), 

% = i2-(r(P-fQ)J 

^'^e'-e '= 2m~ <^>'- 

^ 3/ is Yonng's modulug, v is Poisson's ratio, n is the constant of rigidity, and 
(m - \n) that of compressibility. 



235 cZ.] CYLINDRICAL SHELL. 403 

The energy to, corresponding to unit of volume, is given by 

+ 2(m-n)(/(7 4-5r6+e/) + n(a^ + 6» + c^) (7). 

In the application to a lamina, supposed parallel to the plane 
xy, we are to take iZ = 0, S = 0, T = 0, so that 

<7 = -a^-^, a = 0, 6 = (8). 

Thus in terms of the extensions e, f, parallel to x, y, and of the 
shear c, we get 

u; = n{e» + /« + '^7''(e+/)= + ic4 (9). 

This is the energy reckoned per unit of volume. In oi"der to 
adapt the expression to our purposes, we must multiply it by the 
thickness (2A). Hence as the energy per unit area of a shell 
of thickness 2h, we may take in the notation adopted at the com- 
mencement of this section. 



2nh^ 






(10). 



This expression may be applied to curved as well as to plane 
plates, for any modification due to curvature must involve higher 
powers of A. The same is true of the energy of bending. 

236 e. We are now prepared for the investigation of the 

extensional vibrations of an infinite cylindrical shell, assumed to 

be periodic with respect both to z and to <f>. It will be convenient 

to denote by single letters the displacements parallel to z, <^, r ; 

we take 

Sz= u, aS^ = v, Sr^tu (1). 

These functions are to be assumed proportional to the sines or 
cosines o{ jz/a and 8<f>, Various combinations may be made, of 
which an example^ is 

u=U cos 8<t> cos jz/a, v = F sin s<f> sin jz/a, 

» 

w== W cos s<l> sin jz/a (2); 

sothat(l), (2), (3),§235d 

a.€i = -'jUcoss<f>smjz/a (3), 

a.€2 = (W + 8V)coss<f> sin jz/a (4), 

a.'Gj = (— sU ■]' jV) sin 8<f> cos jz/a (5). 

^ Additions of Jr to «0, or to jz/a, or to both, may of course be made at pleasure. 

26—2 



404 CURVED PLATES OR SHELLS. [235 6. 

The potential energy per unit area is thus (10) § 235 d 
2nha--^lcos^s<f>8m^jz/aijW^-^(W + 8Vy'^^^^ 

+ i8in«^cos«>/a(-«Cr + jF)*l (6). 

Again, if p be the volume density, the kinetic energy per unit 
of area is 

ph\ [--Tr) cos' «<^ cos' j>/a + f-jT-) sin'^i^sin'j-^/a 

-^ j cos' 8^ sin' j^/a (7). 

In the integration of (6), (7) with respect to z and ^, J is the 
mean value of the square of each sine or cosine.^ We may then 
apply Lagrange's method, regarding U, V, W as independent 
generalized co-ordinates. If the type of vibration be cospt, 
and p^p/n = k^, the resulting equations may be written 

{2(i\r+l)j' + «'-ifc'a'} Cr-(2i\r+l)>F-2i^jTf = 0...(8), 

-(2JV'+ l)>Cr+ { j'+ 2(iV^+ ly- A;'a'} F+ 2(i\r+ l)«Tr = 0...(9), 

-2i«5cr+2(i\r+i)5F+{2(i\r+i)-jfc'a'|Tr=o...(io), 

where JVr = ^_r_? (H), 

The frequency equation is that expressing the evanescence of 
the determinant of this triad of equations. On reduction it may 
be written 

[Jfa' - i^ - a*] [i^a^ [Ar^a' - 2 (iV + 1) ( j' + s* + 1)] 

+ 4(2i\r+ 1) j'} + 4 (2iV^+ l)jV = (12).' 

These equations include of course the theory of the extensional 
vibrations of a plane plate, for which a = oo . In this application 
it is convenient to write o^ = y, s/a = ^, j/a = 7. The displace- 
ments are then 

w= Ucosfiycosyz, v = V sin jSy sin yz, w = W cos jSt/sinyz 

...(13). 

^ In the physical problem of a simple cylinder the range of integration for ^ ia 
from to 2t ; but mathematically we are not confined to one revolution. We may 
conceive the sheU to consist of several superposed convolutions, and then s is not 
limited to be a whole number. 

' Note on the Free Vibrations of an infinitely long Cylindrical Shell. Proe, 
Roy. Soc., vol. 45, p. 446, 1S89. 



235 e.J PLANE PLATE. 405 

When a is made infinite while ^, 7 remain constant, the 
equations (10), (8), (9) ultimately assume the form Tr= 0, and 

{2(iV+l)7« + ^-ifc»}Cr-(2iV'+l)7/3F=0...(14), 

- {2N + 1) 7)8Cr -h {7« + 2 {N+ 1) /8« - *»} F- .. .(15) ; 

and the determinantal equation (12) becomes 

A;«[jfca«^-.y3a][i2^2(iV+l)(7' + )8«)] = (16). 

In (16), as was to be expected, h^ appears as a function of 
()8* + y). The first root A:* = relates to flexural vibrations, 
not here regarded. The second root is 

h^^^ + rf (17), 

or p'^-A^-^i") (18). 

At the same time (14) gives 

ryCr-)8F=0 (19). 

These vibrations involve only a shearing of the plate in its own 
plane. For example, if 7 = 0, the vibration may be repre- 
sented by 

n = cos ^y cos p^, vssO, w^O (20). 

The third root of (16) 

A:> = 2(i\r+l)(^ + 7') = -^(/3» + 7») (21) 

gives »2= '- '- (22). 

° ^ m-^n p ^ ^ 

The corresponding relation between U and F is 

;3Cr+7F=0 (23). 

A simple example of this case is given by supposing in (13), 
(23), /3 = 0. We may take 

w = cos72r cosp^, t; = 0, w^O (24), 

the motion being in one dimension. 

Reverting to the cylinder we will consider in detail a few 
particular cases of importance. The first arises when j = 0, that is, 
when the vibrations are independent of z. The three equations 
(8), (9), (10) then reduce to 

(«2-A»aOI7=0 (25), 

{2(i\r-hl)«^-A;*a»}F + 2(iV+l)«Tf = (26), 

2(iV+l)«F+{2(J^ + l)-ifc»a«}Tr=0 (27); 



406 CURVED PLATES OR SHELLS. [235 6. 

and they may be satisfied in two ways. Fii-st let F= F'= ; then 
U may be finite, provided 

^-h^a^ = (28). 

The corresponding type for u is 

u =coss(l> COS pt (29), 

where p^ = — ^ (30). 

In this motion the material is sheared without dilatation of area 
or volume, every generating line of the cylinder moving along 
its own length. The frequency depends upon the cii*cumferential 
wave-length, and not upon the curvature of the cylinder. 

The second kind of vibrations are those for which C/' = 0, so 
that the motion is strictly in two dimensions. The elimination of 
the ratio V/ W from (26), (27) gives 

A;2a2{i«a^-2(JVr+l)(l+5»)} = (31), 

as the frequency equation. The first root is A:* = 0, indicating 
infinitely slow motion. The modes in question are flexural, for 
which, according to our present reckoning, the potential energy 
is evanescent. The corresponding relation between V and W is 
by (26) 

«F+F = (32), 

giving in (3), (4), (5), 

€i = 0, €2=0, w = 0. 
The other root of (31) is 

Ar*a* = 2(i\r-hl)(l+s2) (33), 

or j9* = — ; (34); 

while the relation between V and W is 

F-«TF = (35). 

The type of the motion may be taken to be 

u = 0, v^^ssinsif) cos pt, w = cos s<f) cos pt (36). 

It will be observed that when 8 is very large, the flexural 
vibrations (32) tend to become exclusively radial, and the exten- 
sional vibrations (35) tend to become exclusively tangential. 



235 e.] EXTENSIONAL VIBRATIONS. 407 

Another important class of vibrations are those which are 
chai*acterized by symmetry round the axis, for which accordingly 
5 = 0. The general frequency equation (12) reduces in this case to 

{i^a«-j»}{Jfc^a»[Jfc^a«--2(iV+l)(j^ + l)] + 4(2iV'-fl)j*} = 

...(37). 

Con*esponding to the first root we have U=0, Tr = 0, as is 
readily proved on reference to the original equations (8), (9), (10) 
with 5=0. The vibrations are the purely torsional ones repre- 
sented by 

t^ = 0, V = sin (jz/a) cos pt, w — (38), 

where !>' = —, (39). 

The frequency depends upon the wave-length parallel to the 
axis, and not upon the radius of the cylinder. 

The remaining roots of (37) correspond to motions for which 
F=0, or which take place in planes passing through the axis. 
The general character of these vibrations may be illustrated by 
the case where j is small, so that the wave-length is a large 
multiple of the radius of the cylinder. We find approximately 
from the quadratic which gives the remaining roots 

t.„..a<!|^n... («,. 

The vibrations of (40) are almost purely radial. If we suppose 
that j actually vanishes, we fall back upon 

A?*a' = 2(i\r+1) (42), 

and p«= --- _ (43)1, 

obtainable from (33), (34) on introduction of the condition « = 0. 
The type of vibration is now 

w = 0, t; = 0, w = cos pt (44). 

^ This equation was first given by Love in a memoir "On the smaU Free 
Vibrations and Deformation of a thin Elastic Shell," Phil, Trans,, vol. 179 (1888), 
p. 628. 



408 CUBVED PLATES OB SHELLS. [235 e. 

On the other hand, the vibrations of (41) are ultimately purely 
axial. As the type we may take 

u = co8Jz/a .C08 pt, v^O, w = -^ — J sin j^/d . cos p^ . . . (45), 

, , 3m — n nj* 
where ©* = —^ (46). 

^ m pa^ ^ ^ 

Now, if } denote Young's modulus, we have, § 214, 

g = n (3m — n)/m, 

sothat p'=^| (47). 

Thus u satisfies the equation 

dhi _ q d*u 

which is the usual formula (§ 150) for the longitudinal vibrations 
of a rod, the fact that the section is here a thin annulus not 
influencing the result to this order of approximation. 

Another particular case worthy of notice arises when 8^1, so 
that (12) assumes the form 

ifc«a«(*'a'-j'-l)[*"a'-2(i«^+l)(j« + 2)] 

+ 4j« (A^a* - j«) (2i\r+ 1) = 0. . .(48). 

As we have already seen, if j be zero, one of the values of k^ 
vanishes. If j be small, the corresponding value of A:* is of the 
order j*. Equation (48) gives in this case 

A'a' = ^^^l^-' (49); 

or in terms of p and q, 



The type of vibration is 

ti = 



^ = 2$ (««)• 



t; = 8in^sinj>/a.cos/}^ I (51), 

w = — cos <f> sin jz/a . cos pt j 

and corresponds to the flexural vibrations of a rod (§ 163). In 
(51) V satisfies the equation 

d^ qa^ ^^ — 
d^'*"2pd^~ ' 



235 6.] EXTENSIONAL VIBRATIONS. 409 

in which ^a^ represents the square of the radius of g}rration of the 
section of the cylindrical shell about a diameter. 

This discussion of particular cases may sufiSce. It is scarcely 
necessary to add, in conclusion, that the most general deformation 
of the middle surface can be expressed by means of a series of such 
as are periodic with respect to z and <^, so that the problem con- 
sidered is really the most general small motion of an infinite 
cylindrical shell 

The extensional vibrations of a cylinder of finite length have 
been considered by Love in his Theory of Elasticity^ (1893), where 
will also be found a full investigation of the general equations of 
extensional deformation. 

236/. When a shell is deformed in such a manner that no 
line traced upon the middle surface changes in length, the term of 
order h disappears from the expression for the potential energy, 
and unless we are content to regard this function as zero, a 
further approximation is necessary. In proceeding to this the 
first remark that occurs is that the quality of inextension attaches 
only to the central lamina. Consider, for example, a portion of a 
cylindrical shell, which is bent so that the original curvature is 
increased. It is evident that while the middle lamina remains 
unextended, those laminae which lie externally must be stretched, 
and those that lie internally must be contracted. The amount of 
these stretchings and contractions is proportional in the first place 
to the distance from the middle surface, and in the second place to 
the change of curvature which the middle surface undergoes. The 
potential energy of bending is thus a question of the curvatures of 
the middle surface. Displacements of translation or rotation, such 
as a rigid body is capable of, may be disregarded. 

In order to take the question in its simplest form, let us refer 
the original surface to the normal and principal tangents at any 
point P as axes of co-ordinates, and let us suppose that after 
deformation the lines in the sheet originally coincident with the 
principal tangents are brought back (if necessary) so as to occupy 
the same positions as at first. The possibility of this will be 
apparent when it is remembered that, in virtue of the inexten- 
sion of the sheet, the angles of intersections of all lines traced 

^ Also Phil. Tratii. vol. 179 a, 1888. 



410 CURVED PLATES OR SHELLS. [235/ 

upon it remain unaltered. The equation of the original surface in 
the neighbourhood of the point being 

.-t(f+f) (1), 

\Pi Pi/ 
that of the deformed surface may be written 

"^ +-i^+2T^l (2). 



-ifcl 



Spi Pi + S/>a 

In strictness (pi -f Spi)~S (p^ + ^Pi)"^ are the curvatures of the 
sections made by the planes a?, y ; but since the principal curvatures 
are a maximum and a minimum, they represent in general with 
sufficient accuracy the new principal curvatures, although these 
are to be found in slightly different planes. The condition of 
inextension shews that points which have the same x, y in (1) 
and (2) are corresponding points ; and by Gauss's theorem it is 
further necessary that 

^^ + ^^ = (3). 

Pi pi 

It thus appears that the energy of bending will depend in 
general upon two quantities, one giving the alterations of principal 
curvature, and the other r depending upon the shift (in the 
material) of the principal planes. 

The case of a spherical surface is in some respects exceptional. 
Previously to the bending there are no planes marked out as 
principal planes, and thus the position of these planes after 
bending is of no consequence. The energy depends only upon 
the alterations of principal curvature, and these by Gauss's theorem 
are equal and opposite, so that, if a denote the radius of the 
sphere, the new principal radii are a + Sp, a — Sp. If the equation 
of the deformed surface be 

2z = Aai' + 2Bxy + Cy- (4). 

we have (a + 8p)~^ + (a — Bp)"^ = il + C, 

so that (S-y^iiA^Cy + B' (5). 

We have now to express the elongations of the various laminae 
of a shell when bent, and we will begin with the case where t = 0, 



235/] ENERGY OF BENDING. 411 

that is, when the principal planes of curvature remain unchanged. 

It is evident that in this case the shear c vanishes, and we have to 

deal only with the elongations e and /parallel to the axes, § 235 d. 

In the section by the plane of zx, let s, s' denote corresponding 

infinitely small arcs of the middle surface and of a lamina distant 

h from it. If -i^ be the angle between the terminal normals, 

8 = p^yjr, s' = {pi -f A) '<^, s' — s = A^/r. In the bending, which leaves 

s unchanged, 

8s' = /tS^ = /i8S(l/pi). 

Hence e = Ss'/s' = A 8 ( 1 /pi), 

and in like manner /= AS(l//}s). Thus for the energy U per unit 
area we have 

[\ pJ \ pJ m + nx pi p^J ) 
and on integration over the whole thickness of the shell (2A) 

U^^^\(Bl.)\(B^)\'!^U^ + B^yl (6). 

3 i\ pJ \ PiJ m + n\ pi pJ ) 

This conclusion may be applied at once, so as to give the result 

applicable to a spherical shell; for, since the original principal 

planes are arbitrary, they can be taken so as to coincide with the 

principal planes after bending. Thus t = 0; and by Gauss's 

theorem 

8(1/^0 + 8 (1/P2) = 0, 

so that Cr = ^'(siy (7), 

where B(l/p) denotes the change of principal curvature. Since 
e = — / 5^ = 0, the various laminae are simply sheared, and that in 
proportion to their distance from the middle surface. The energy 
is thus a function of the constant of rigidity only. 

The result (6) is applicable directly to the plane plate; but 
this case is peculiar in that, on account of the infinitude of pi, p^ 
(3) is satisfied without any relation between Spi and Sp^. Thus for 
a plane plate 

2nhUl 1 m-w/1 IV) /ox 

where l/pi, l/pa, are the two independent principal curvatures after 
bending ^ 

1 This will be fonnd to agree with the valae (2) § 214, expressed in a different 
notation. 



u= 



412 CURVED PLATES OR SHELLS, [235/. 

We have thus far considered r to vanish; and it remains to 
investigate the effect of the deformations expressed by 

S^ = Ta:y = iT(f«-i7») (9), 

where f , tf relate to new axes inclined at 45 "" to those of x, y. The 
curvatures defined by (9) are in the planes of f, i;, and are equal 
in numerical value and opposite in sign. The elongations in these 
directions for any lamina within the thickness of the shell are hr^ 
- At, and the corresponding energy (as in the case of the sphere 
just considered) takes the form 

U' = ^ (10). 

This energy is to be added* to that already found in (6); and 
we get finally 

2nA» 



U^ 



as the complete expression of the energy, when the deformation is 
such that the middle surface is unextended. We may interpret t 
by means of the angle x* through which the principal planes are 
shifted; thus 

^ = 2x(J-7) (12)- 

236 g. We will now proceed with the calculation of the 
potential energy involved in the bending of a cylindrical shell. 
The problem before us is the expression of the changes of prin- 
cipal curvature and the shifts of the principal planes at any point 
P {z, <f>) of the cylinder in terms of the displacements u, v,w. As in 
§ 235 /, take as fixed co-ordinate axes the principal tangents and 
normal to the undisturbed cylinder at the point P, the axis of x 
being parallel to that of the cylinder, that of y tangential to the 
circular section, and that of ^ normal, measured inwards. If, as it 
will be convenient to do, we measure z and <f> from the point P, we 
may express the undisturbed co-ordinates of a material point Q in 
the neighbourhood of P, by 

x = z, y = a<f>, ^ = ia<^' (1). 

^ There are clearly no terms involving the prodnots of r with the changes of 
principal curvature S{p^~^), B{pf^); for a change in the sign of r can have no 
influence upon the energy of the deformation defined by (2). 



2355^.] APPLICATION TO CYLINDER. 413 

During the displacement the co-ordinates of Q will receive the 

increments 

u, W7 sin ^ + 1; cos ^, —w cos ^ + v sin (f) ; 

so that after displacement 

x^zz-hu, y — a4>-\- w<f> + v (1 — J^*), 

or, if u, V, w be expanded in powers of the small quantities z, <f>, 

du du , 

i J dv dv , 
y = a<f> + vJ,<f> + v, + ^^z + ^^^4>+ (3). 

dv , dv ., 

+ dF/*-^#/' W' 

Wo» Vo> . • • being the values of t^, v at the point P. 

These equations give the co-ordinates of the various points of 
the deformed sheet. We have now to suppose the sheet moved as 
a rigid body so as to restore the position (as far as the first power 
of small quantities is concerned) of points infinitely near P, A 
purely translatory motion by which the displaced P is brought 
back to its original position will be expressed by the simple 
omission in (2), (3), (4) of the terms Uq, v©, Wq respectively, which 
are independent of z, <f>. The eflfect of an arbitrary rotation is 
represented by the additions to x, y, ^ respectively of ya^ — fo),, 
fwi — a?ft)3, xa)^ — ycoi ; where for the present purpose a>i, cd^, 0)3 are 
small quantities of the order of the deformation, the square of 
which is to be neglected throughout. If we make these additions 
to (2), &c., substituting for x, y, f in the terms containing their 
approximate values, we find so far as the first powers of z, <\> 

du du , , , 
. , , dv , dv , 

^, dw dw . ^ . , , 



414 



CURVED PLATES OR SHELLS. 



[235 gr. 



Now, since the sheet is assumed to be unextended, it must be 

possible so to determine cdi, «2» <^s that to this order x^z, y = a<f>, 

f=0. Hence 

du ^ 

+ acDs = 0, 



du 
dzo 



dv ^ 



d<f>c 

dv ^ 
i^oH-:r. =0, 

dw - 

-rr "" ^0 + OKi)i = 0. 
(i<f>o 



The conditions of inex tension are thus (if we drop the suflBces 
as no longer required) 



dz • 



dv - 



du ^^ _ A 
d(f) dz 



(5), 



which agree with (8) § 235 c. 

Returning to (2), &c., as modified by the addition of the trans- 
latory and rotatory terms, we get 

x^z+ terms of 2nd order in z, <f>, 



dHv 



? = ia<^^ + i«;o<A'-i^^^»- 



dy 

dZf4<f>t 



z<f> 



, d^w ,^ dv . dv ,^ 
or since by (5) d^w/dz^ = 0, and dv/d<f> = — w. 



dHv 



dHu 



dv 



The equation of the deformed surface after transference is thus 



^^xy 



[I dv 1 d'w 1 



r+y' 



2a ■■2a^^^ 



a dzo a dZi^(}>Q 
Comparing with (2) § 235/ we see that 



1^ dhu) 
2a* d^o'J 



. . . . (6). 



8- = 0, 
Pi 



so that by (11) § 235/ 



.1 \ ( ^d'w\ 



1 (dv dhu \ 



d<f>y' 



_ 1 Idv a*w \ ^. 
~a[di~d^)"'^'^' 



„_ 4wA« ( m 1 / d'wy (dv d'^w V) 
~ 3a^ [m + n a* V dtfy') "*" U^ d^) J ^ ^' 



235 gr.] APPLICATION TO CYLINDER. 415 

This is the potential energy of bending reckoned per unit of 
area. It can, if desired, be expressed by (5) entirely in terms of v^. 

We will now apply (8) to calculate the whole potential energy 
of a complete cylinder, bounded by plane edges z=±l, and of 
thickness which, if variable at all, is a function of z only. Since 
u, V, w are periodic when (f) increases by 27r, their most general 
expression in accordance with (5) is [compare (10), &c., § 235 c] 

v = 2 [(Aga H- Bgz) cos «<^ — (^/a H- B/z) sin 8(f>] (9), 

w = X[8 (Aga + Bgz) sin s<f>-\-s (Aga H- B/z) cos s<f>] .... (10), 

i^ = 2 [— s^^Bga sin 8<t> — s''^B/a cos «0] (11), 

in which the summation extends to all integral values of 8 from 
to 00 . But the displacements corresponding to 5 = 0, 8=1 are 
such as a rigid body might undergo, and involve no absorption of 
energy. When the values of u, v, w are substituted in (8) all the 
terras containing products of sines or cosines with different values 
of 8 vanish in the integration with respect to <f>, as do also those 
which contain cos 8<f> sin 8(f>, Accordingly 

Jo ^ 3a l_m + n a* ^ 

{{Aga + Bgzy + (Ag'a + B/zf] + 2 (^ - 1)^ (5.' + B/')\ . . .(12). 

Thus far we might consider A to be a function of z ; but we will 
now treat it as a constant. In the integration with respect to z 
the odd powers of z will disappear, and we get as the energy of the 
whole cylinder of radius a, length 21, and thickness 2A, 

+ ^^(5,^H-5/o}+5,^ + 5/^] (13), 

in which 5=2, 3, 4,.... 

The expression (13) for the potential energy suffices for the 
solution of statical problems. As an example we will suppose 
that the cylinder is compressed along a diameter by equal forces 
F, applied at the points z = Zi, <^ = 0, <f>^ir, although it is true 
that so highly localised a force hardly comes within the scope of 

^ From the general equations of Mr Love's memoir already cited a concordant 
result may be obtained on introduction of the appropriate conditions. 



416 CURVED PLATES OR SHELLS. [235 ^r. 

the investigation, in consequence of the stretchings of the middle 
surface, which will occur in the immediate neighbourhood of the 
points of application ^ 

The work done upon the cylinder by the forces F during the 
hypothetical displacement indicated by hA,, &c., will be by (10) 

- FSj? {ahAJ + z,iB,') (1 + cos w), 
so that the equations of equilibrium are 

^^"=0, 4 = 0. 



dAg * dBg 

T . / = - (1 + cos sw) saF, j^, = - (1 4- cos sir) sziF. 
Thus for all values of 8, 

and for odd values of s, Ag = B, = 0. 
But when 8 is even, 

m + u • SimhHis'-ir ^^*^' 

(m + n3a«"^ I • "" SirnhH {i^ -^ ly ^^'^^• 

and the displacement w at any point {z, (f>) is given by 

w = 2 (A::a + B^z) cos 2<^ + 4 {A^a + B^z) cos 4<^ + . . .(16), 
where A^y B./, -4/,... are detennined by (14), (15). 

A further discussion of this solution >vill be found in the 
memoir^ from which the preceding results have been taken. 

We will now proceed with the calculation for the firequencies 
of vibration of the complete cylindrical shell of length 21. If the 
volume-density* be p, we have as the expression of the kinetic 
energy by means of (9), (10), (11), 

T = i . 2Ap . jj(u' + v- + w^) ad<f>dz 

= 27rphla S [a^ (1 + ^) (i, + A/^) 

+ W (1 + «0 + s-^a'] (B,' -h 5/0} (17). 

^ Whatever the curvature of the surface, an area upon it may be taken so small 
as to behave like a plane, and therefore bend, in violation of Gauss's condition, 
when Eubjected to a force which is so nearly discontinuous that it varies sensibly 
within the area. 

» Proc, Roy, Soc. vol 45, p. 105, 1888. 

3 This can scarcely be confused with the notation for the curvature in the 
preceding parts of the investigation. 



235 gr.] FREQUENCY EQUATION. 417 

From the expressions for V and T in (13), (17) the types and 
frequencies of vibration can be at once deduced. The fact that 
the squares, and not the products, of A,, B,, are involved, shews 
that these quantities are really the normal co-ordinates of the 
vibrating system. If -4„ or A/, vary as cosp^, we have 

^' *m-hn/>a* ^ + 1 ^^^^• 

This is the equation for the frequencies of vibration in two 
dimensions, § 233. For a given material, the frequency is pro- 
portional directly to the thickness and inversely to the square 
on the diameter of the cylinder^ 

In like manner if £,, or B,\ vary as coap,% we find 

1 3a'mH-n 
, mn A« (^-g)' 71' m 

P' '^m-^-npa' «» + l 3a« " ^^^* 

(«* + «»)> 

If the cylinder be at all long in proportion to its diameter, the 
difference between p, and p, becomes very small. Approximately 
in this case 

l>./i'. = l + 2^,(--^- -^ (20); 

or, if we take m = 2n, s = 2, 

236 A. We now pass on to the consideration of spherical 
sheila The general theory of the extensional vibrations of a 
complete shell has been given by Lamb*, but as the subject is 
of small importance from an acoustical point of view, we shall 
limit our investigation to the very simple case of symmetrical 
radial vibrations. 

If w be the normal displacement, the lengths of all lines upon 
the middle surface are altered in the ratio (a + w): a. In calcu- 
lating the potential energy we may take in (10) § 235 d 

€i = €a = w/a, w = ; 

^ There is nothing in these laws special to the cylinder. In the case of similar 
shells of any form, vibrating by pore bending, the frequency will be as the thick- 
nesses and inversely as corresponding areas. If the similarity extend also to the 
thickness, then the frequency is inversely as the linear dimension, in accordance 
with the general law of Canchy. 

' Proc. Lond, Math. 8oe, xiv. p. 50, 18S2. 

R. 27 



418 CURVED PLATES OR SHELLS. [235 k. 

80 that the energy per unit area is 



. , 3m — n w;^ 

4ink ; r, 

m + n or 



or for the whole sphere 



F=4^a».'4nA (1). 

Also for the kinetic energy, if p denote the volume density, 

r=i.47ra«.2A.p.t(;' (2). 

Accordingly if w = TT cos pt, we have 

4yi 3m-n 
^ "aV m + n ^^^' 

as the equation for the frequency (p/^ir). 

As regards the general theory Prof Lamb thus summarizes his 
results. "The fundamental modes of vibration fall into two 
classes. In the modes of the First Class, the motion at every 
point of the shell is wholly tangential. In the nth species of 
this class, the lines of motion are the contour lines of a surface 
harmonic Sn (Ch. xvii.), and the amplitude of vibration at any 
point is proportional to the value of dSn/de, where de is the angle 
subtended at the centre by a linear element drawn on the surface 
of the shell at right angles to the contour line passing through the 
point. The frequency {pl^ir) is determined by the equation 

k»a^ = (M-l)(ri + 2) (i), 

where a is the radius of the shell, and k^=p^pjn, if p denote the 
density, and n the rigidity, of the substance." 

" In the vibrations of the Second Class, the motion is partly 
radial and partly tangential. In the nth species of this class the 
amplitude of the radial component is proportional to Sn,Si surface 
harmonic of order n. The tangential component is everywhere at 
right angles to the contour lines of the harmonic Sn on the surface 
of the shell, and its amplitude is proportional to AdSn/de, where 
A is a certain constant, and de has the same meaning as before." 
Prof. Lamb finds 



A 



k'a'-4rf 
2^(71+1)7 ^^^^' 



where k retains its former meaning, and 7 = (1 + <r)/(l — a), a 



235 h.] 



COMPLETE SPHERE. 



419 



denoting Poisson's ratio. " Corresponding to each value of n there 
are two values of k*a*, given by the equation 

k*a* - k=*a« {(n« + n + 4)7 + n» + n- 2} +4 (71=* + n- 2) 7 = 0...(iii). 

Of the two roots of this equation, one is < and the other > 4sy. It 
appears, then, from (ii) that the cofresponding fundamental modes 
are of quite different characters. The mode corresponding to the 
lower root is always the more important." 

" When n = 1, the values of k'^'a^ are and 67. The zero root 
corresponds to a motion of translation of the shell as a whole 
parallel to the axis of the harmonic Si. In the other mode the 
radial motion is proportional to cos 0, where is the co-latitude 
measured from the pole of Si ; the tangential motion is along the 
meridian, and its amplitude (measured in the direction of in- 
creasing) is proportional to ^ sin 0J* 

*' When n = 2, the values of ka corresponding to various values 
of a- are given by the following table : — 



a = 


cr = i 


c- = fV 


-i 


cr = i 

1-215 
5-703 


M20 
3-570 


1-176 
4-391 


M85 
4-601 


1-190 
4-752 



The most interesting variety is that of the zonal harmonic. 
Making jS=^(3cos^^— 1), we see that the polar diameter of 
the shell alternately elongates and contracts, whilst the equator 
simultaneously contracts and expands respectively. In the mode 
con-esponding to the lower root, the tangential motion is towards 
the poles when the polar diameter is lengthening, and vice versd. 
The reverse is the case in the other mode. We can hence under- 
stand the great difference in fi-equency." 

Prof. Lamb calculates that a thin glass globe 20 cm. in 
diameter should, in its gravest mode, make about 5350 vibrations 
per second. 

As regards inextensional modes, their form has already been 
determined, (39) &c. § 235 c, at least for the case where the 
bounding curve and the thickness are symmetrical with respect 
to an axis, and it will further appear in the course of the present 
investigation. What remains to be effected is the calculation of 

27—2 



420 CURVED PLATES OB SHELLS. [235 h. 

the potential energy of bending corresponding thereto, as depend- 
ent upon the alterations of curvature of the middle surface. The 
process is similar to that followed in § 235^ for the case of the 
cylinder, and consists in finding the equation of the deformed 
surface when referred to rectangular axes in and perpendicular 
to the original surface. 

The two systems of co-ordinates to be connected are the usual 
polar co-ordinates r, 0, <f>, and rectangular co-ordinates x, y, f, 
measured from the point P, or (a, 0o, ^), on the undisturbed 
sphere. Of these x is measured along the tangent to the 
meridian, y along the tangent to the circle of latitude, and (f 
along the normal inwards. 

Since the origin of ^ is arbitrary, we may take it so that 
^0 = 0. The relation between the two systems is then 

X == r {"8X11(0 — 00)-^ sin cos 0o(l — cos <t>)} (4), 

y^rsindsin^ (5), 

f = — r {cos (0 — ^o) — sin 0o sin ^ (1 — cos <^)} -h a . . . .(6). 

If we suppose r^a, these equations give the rectangular 
co-oi-dinates of the point (a, 0, <^) on the undisturbed sphere. 
We have next to imagine this point displaced so that its polar 
co-ordinates become a + Sr, 0'\-h0, <^ + &^, and to substitute these 
values in (4), (5), (6), retaining only the first power of Sr, S0, 80. 
Thus 

a? = (a H- Sr) {— sin {0 — ^o) + si^i cos 0^ (1 — cos <f>)} 
H- aS0 {— cos (^ — ^o) + cos ^ cos ^o (1 — cos <^)} 
-f aS<^sin^cos^osin<^ (7), 

y s (a + Br) sin d sin (^ 

+ aS0 cos sin <f> + aS<f> sin cos <f> (8), 

^^a-(a + &r) {cos {0 - ^o) - sin 0o sin ^ (1 - cos 0)} 

H- aS0 {sin (0 - ^o) + sin ^o cos ^ (1 — cos <^)} 

+ aS<^ sin ^0 sin 0sin<f> (9). 

These equations give the co-ordinates of any point Q of the sphere 
after displacement ; but we shall only need to apply them in the 
case where Q is in the neighbourhood of P, or (a, 00, 0), and then 
the higher powers of (0 — ^o) and <^ may be neglected. 



— a i 



235 A.] INEXTENSIONAL MODES. 421 

In pursuance of our plan we have now to imagine the displaced 
and deformed sphere to be brought back as a rigid body so that 
the parts about P occupy as nearly as possible their former 
positions. We are thus in the first place to omit from (7), (8), 
(9) the terms (involving S) which are independent of {6 — 0o), <f>» 
Further we must add to each equation respectively the terms 
which represent an arbitrary rotation, viz. 

y ©8 — {©a , f 0)1 — a:G)8 , aeo^ — y ©i , 

determining (»i, o),, 0)3 in such a manner that, so far as the first 
powers of {0 — ^0), <f>, there shall be coincidence between the original 
and displaced positions of the point Q. 

If we omit all terms of the second order in (^ — ^0) and <f>, we 
get from (7) Ac. 

oo = -a(0-0o)-Bro{O'-0o) 

[S0o]'\'^{0-0o)-¥j^<f>\ + aS<{>oSm0oCos0o.<l>... (10), 
y = a sin ^0 • <^ + Sro sin ^0 • <^ + (^^0o cos 0o . (f) 

+ aB<f>oCOB0o{0-0o) (11), 

f=t-Ki-g(*-«.)-^* 

+ aB0o{0-0o) + aB(t>oSm^0o.<f> (12), 

Bvq &c. representing the values appropriate to P, where (0 — ^0) 
and (}> vanish. The translation of the deformed surface necessary 
to bring P back to its original position is represented by the 
omission of the terms included in square brackets. The arbitrary 
rotation is represented by the additions respectively of 

a sin ^0 • <A • ®8> a (^ — ^0) «8, - a (^ - ^0) Wa — ct sin ^0 • <^ • Wi 5 
and thus the destruction of the terms of the first order requires 
that 

Srla + dB0/d0^O (13), 

- d8^/d<^ + sin ^ cos ^ S<^ H-sin ^ ©8 = (14); 

sin ^ dS<^/d^ + cos ^ S<^ + 0)8 = (15), 

(Sr/a) sin ^ + 8^ cos ^ + sin ^ dS^/(ii^ = (16); 

-d8(r/a)/d^ + S^-o), = (17), 

-dS(r/a)/d<^ + 8in»^S<^- sin ^0)1=0 (18); 

the suffixes being omitted. 



422 CURVED PLATES OR SHELLS. [235 h. 

These six equations determine (o^^ o),, o),, giving as the three 
conditions of inextension 

Sr/a + dh0/d0^O (19), 

dS0/dit> + sin^ dB4>/d0 == (20), 

Sr/a + cot ^ S^ + d8<^/d<^ = (21). 

From (19), (20), (21), by elimination of Sr, 

or, since sin d/d0 = d/d log tan ^, 

d<^Une/ dlogtani^ " ^^*^' 

d4> dlogtani^Vsin^y ^"^* 

From (24), (25) we see that both S^ and B0/am0 satisfy an 
equation of the second order of the same form, viz. 

d^u d*u __ 
d (log tan ^0y "^ d^« " " ^^^^• 

If the material system be symmetrical about the axis, tt is a 
periodic function of (f>, and can be expanded by Fourier's theorem 
in a series of sines and cosines of (f> and its multiples. Moreover 
each term of the series must satisfy the equations independently. 
Thus, if u varies as cos «0, (26) becomes 

diloglw -"'^-^ (27); 

whence u = ^' tan' ^^ + 5^ cot* ^^ (28), 

where A' and R are independent of 0. If we take 

S<^ = cos «<^ [il, tan' i^ + 5, cot' i^] (29), 

we get for the corresponding value of S0 from (24) 

S^/sin^ = -sin«<^[il,tan'i^-5,cot'i^] (30); 

and thence from (21) 
&r/a = sin 8<{> [A, (s + cos 0) tan' ^0 + 3,(8- cos 0) cot' ^0] . . .(31), 

as in (39), (40), (41) § 235 c. 



235 ^.] INEXTENSIONAL MODES. 423 

The second solution (in Bg) may be derived from the first (in Ag) 
in two ways which are both worthy of notice. The manner of deri- 
vation from (27) shews that it is sufficient to alter the sign of 8, 
tan*^^ becoming cot'^O, sin«<^ becoming — sin 8(f>, while cos 8<f> 
remains unchanged. The other method depends upon the con- 
sideration that the general solution must be similarly related to 
the two poles. It is thus legitimate to alter the first solution by 
writing throughout (tt — 0) in place of 0, changing at the same 
time the sign of S0, 

If we suppose « = 1, we get 

sin ^S^ = cos <^ [ill -h -Bi — (-^i — A) cos 0], 

8^ = - sin ^ [^1 - 5i - (A + A) cos ^], 

Sr/a = sin <^ [(A^ + B^) sin 0], 

The displacement proportional to (Ai — Bi) is a rotation of the 
whole surface as a rigid body round the axis = ^, <^ = ; and 
that proportional to (Ai + Bi) represents a translation parallel to 
the axis 0^^, <^ = ^7r. The complementary translation and 
rotation with respect to these axes is obtained by substituting 
<^ H- ^TT for <^. 

The two other motions possible without bending correspond to 
a zero value of 8, and are readily obtained from the original 
equations (19), (20), (21). They are a rotation round the axis 
^ = 0, represented by 

B0 = 0, S<^ = const., Sr = 0, 
and a displacement parallel to the same axis represented by 

or S<^ = 0, S^s=78in^, Sr = ^ya cos 0, 

If the sphere be complete, the displacements just considered, 
and corresponding to « = 0, 1, are the only ones possible. For 
higher values of 8 we see from (31) that Sr is infinite at one or 
other pole, unless Ag and Bg both vanish. In accordance with 
Jellet's theorem' the complete sphere is incapable of bending. 

If neither pole be included in the actual surface, which for 
example we may suppose bounded by circles of latitude, finite 

1 *<0n the Properties of Inextensible Surfaces," Iriih Acad, Trant., vol. 22, 
p. 179. 1S66. 



424 CURVED PLATES OR SHELLS. [235 A. 

values of both A and B are admissible, and therefore necessary for 
a complete solution of the problem. But if, as would more often 
happen, one of the poles, say ^ = 0, is included, the constants B 
must be considered to vanish. Under these circumstances the 
solution is 

S^ = Ag tan' ^d cos 8<f> \ 

S^ = - ^, sin ^ tan* i^ sin «<^ I (32), 

Sr = -4,a (« + cos 0) tan* \0 sin «<^ J 

to which is to be added that obtained by writing 8<^ + ^tt for «<^, 
and changing the arbitraiy constant 

From (32) we see that, along those meridians for which 
sin 8if> =■ 0, the displacement is tangential and in longitude only, 
while along the intermediate meridians for which cos 8^ = 0, there 
is no displacement in longitude, but one in latitude, and one 
normal to the surface of the sphere. 

Along the equator = ^tt, 

S<^ = Af cos «<^, Z0 = '-Ag sin «<^, irfa = Ag8 sin «0, 

so that the maximum displacements in latitude and longitude are 
equal. 

Reverting now to the expressions for x, y, 5' in (7), (8), (9), 
with the addition of the translatory and rotatory terms by which 
the deformed sphere is brought back as nearly as possible to its 
original position, we know that so far as the terms of the first 
order in {0 — ^o) and <^ are concerned, they are reduced to 

a; = - a (^ - ^o), y = a sin 0^.4>, (;= (33). 

These approximations will suffice for the values of x and y ; but 
in the case of ^ we require the expression complete so as to 
include all terms of the second order. The calculation is straight- 
forward. For any displacement such as Sr in (9) we write 

The additional rotatoiy terms are by (17), (18) 

^ r» —adeji^y io sin d.d^r '"" ^'^ 



235 h.] POTENTIAL ENERGY OF BENDING. 425 

In these we are to retain only those terms in x, y, which are of the 
second order and independent of S, so that we may write 

X = ^a<f>^ sin ^o cos Oq, y = a (^ - ^o) <t> cos ^o- 

In the complete expression for ^ as a quadratic function of 
{0 — ^o) and (f) thus obtained, we substitute x and y from (33). 
The final equation to the deformed sur£Etce, after simplification by 
the aid of (19), (20), (21), may be written 

^^ r Sr_ld«Sr) xy ( 1 d?hr cot^dgr] 
^ 2a| a a d^j asin^l adOd^^ a d^] 

y]; f '_ Sr _ cot^ d8r _ 1 d^h-} 
■^2af a a d0 asm^d d4>^\ ^•**^' 

the suffixes being now unnecessary. 

Taking the value of Br/a from (32) we get 

Sr Id^Sr «»-5 
j^ = — -r-7-2, -dg tan* *^ sm «6 (3o), 

1 d^hr cos^ dlr «*-« - . ..^ . ,«^. 

^/,:j>jjrTH • o/i"Ti =" .-rz-4»^i^ i^cos«6....(36), . 

asm0d0d<f> a sin^ d<f> sm**^ * r \ /> 

Sr cot^dSr 1 d^Sr s^-8 . ^ ,,^ . . ,^,_. 

-j^ r-r-a v::- = -r-r-^-4gtanH^sm«6...(37). 

a a d0 a sm» ^ d<f>^ sin* ^ -* ir \ / 

To obtain the more complete solution corresponding to (31), we 
have only to add new terms, multiplied by Bg, and derived from 
the above by changing the sign of 8, As was to be expected, the 
values in (35) and (37) are equal and opposite. 

Introducing the values now found into (5) § 235 /, we obtain 
as the square of the change of principal curvature at any point 

(S -X = ~^^l [AJ" tan* i0 + B,^ cot» ^0 - 2A,B, cos 28<t>} . . .(38). 

It should be remarked that, if either A, or Bg vanish, (38) is 
independent of <^, so that the change of principal curvature is the 
same for all points on a circle of latitude, and that in any case 
(38) becomes independent of the product AgBg after integration 
round the circumference. The change of curvature vanishes if 
8 = 0fOx; 8=^1, the displacement being that of which a rigid body 
is capable. 

Equations (35) &c. shew that along the meridians where S^ 
vanishes (cos«^ = 0) the principal planes of curvature are the 



426 CURVED PLATES OR SHELLS. [235 h. 

meridian and its perpendicular, while along the meridians where 
Sr vanishes, the principal planes are inclined to the meridian at 
angles of 45°. 

The value of the square of the change of curvature obtained in 
(38) corresponds to that assumed for the displacements in (29) &c., 
and for some purposes needs to be generalised. We may add 
terms with coefficients A,' and B, corresponding to a change 
of 8<^ to («<^ + i7r), and there is further to be considered the 
summation with respect to 8. Putting for brevity t in place of 
tan^d, we may take as the complete expression for [S (!//>)]', 

\^^S-0 ^^"^'^ "^ ^'^'^ ^'"^ ** "^ ^^'^ ■*■ ^''^'^ ^'"^ (** "^ ^""^^1' 
+ [S -^^^ [{A,i^ - Bst-*) cos «<^ + (il.V - 5/e-) cos {84> + \iryX . 

When this is integrated with respect to <j> round the entire 
circumference, all products of the generalised co-ordinates J.«, B^, 
Ai, Bi disappear, so that (7) § 235/ we have as the expression for 
the potential energy of the surface included between two paitiUels 
of latitude 

F = 27r2 (j9» - 8f (h 8in-» {{A,' + A,'') t^ 

+ (B,' + B,'')t-^}d0 (39), 

where H^^nh^ (40). 

In the following applications to spherical surfeu^es where the 
pole ^ = is included, we may omit the terms in B\ and, if 
the thickness be constant, H may be removed from under the 
integral sign. We have 

d0^^.. sin^= 2^ 



so that 

fit / A^*— « 9/2» ^+2 \ 

j^sin- dfdd = i j (1 + e')« «»-« «i<« = i (-' -J- + fi- + i_J. . .(41^. 

In the case of the hemisphere t = 1, and (41) assumes the value 

2«»-l 



,(42). 



4(s»-«) 

Hence for a hemisphere of uniform thickness 

F= \irHt («• - «) (2«> - 1) (4,> + A:^) (43). 



235 /i.] STATICAL PROBLEMS. 427 

If the extreme value of ^ be 60°, instead of 90°, we get in 
place of (42) 

4.3'+»(5»-5) ^ '' 

and V = iirHX 3-<*+») (s' - a) {8^ + 4« - 3) (A,^ 4- ^;«). . .(45). 

These expressions for F, in conjunction with (32), are sufficient 
for the solution of statical problems, relative to the deformation of 
infinitely thin spherical shells under the action of given impressed 
forces. Suppose, for example, that a string of tension F connects 
the opposite points on the edge of a hemisphere, represented by 
^ = i^> <^ = ^TT or |7r, and that it is required to find the deforma- 
tion. It is evident from (32) that all the quantities A/ vanish, 
and that the work done by the impressed forces, coiTCsponding to 
the deformation BA,, is 

— SAgds (sin i^sir + sin fwr} F. 
If 8 be odd this vanishes, and if s be even it is equal to 

— 2BAsa8 sin ^stt.F. 
Hence if s be odd Ag vanishes ; and by (43), if a be even, 

dF/dil, = 7rir(a'-«)(25*-l)^, = -2(Wsini«7r.F; 
whence ^ - - 2a^sin jsir 

By (46) and (32) the deformation is completely determined. 

If, to take a case in which the force is tangential, we suppose 
that the hemisphere rests upon its pole with its edge horizontal, 
and that a rod of weight W is laid symmetrically along the 
diameter O^^w, we find in like manner 

^•"•7rir(*^-«)(2«^-.l) ^*^^ 

for all even values of «, and -4, = for all odd values of a. 

We now proceed to evaluate the kinetic energy as defined by 
the formula 

in which <r denotes the surfeu^e density, supposed to be uniform. 



428 CURVED PLATES OR SHELLS. [235 A. 

If we take the complete value of S(f> from (29), as supplemented by 
the terms in A,\ B,, we have 

J" = 2 [cos 8<f> (A,tf + B,tr*) + cos («<^ + ^tt) (i/t* + £/«-•)]. 

When this expression is squared and integrated with respect 
to (f> round the entire circumference, all products of letters with a 
different su£Sx, and all products of dashed and undashed letters 
even with the same su£Sx, will disappear. Hence replacing cos* 8<l> 
&c. by the mean value i, we may take 



sm 



• • 



in» (^y = i sin» ^ 2 (A.* + i;«) e« 

+ i sin^ X {B,' + B;«) ir^ + sin« 1 {A,B, + A^B.y 

The mean value (30) of (dB0ldty is the same as that just 
written with the substitution throughout of — £ for B, so that we 
may take 

+ sin»^2(^,« + ^;»)r" (49), 

as the mean available for our present purpose. In (49) the 
products of the symbols have disappeared, and if the expression 
for the kinetic energy were as yet fully formed, the co-ordinates 
would be shewn to be normcU. But we have still to include that 
part of the kinetic energy dependent upon dBr/dt As the mean 
value, applicable for our purpose, we have from (31) 

( f£f = i 2 (i.» + i;«) (« + cos ey p 

+ i 2 (£,» + B,'*) (8 - COS eyir^ 

+ 2 (A,B. + A.'B,') («» - COS' ^) (50). 

The expressions (49) and (oO) have now to be added. If we set 
for brevity 



/' 



tan" i0 {(8 + cos «)« + 2 sin'' 0} sin 0d0^/{8) (51), 



or putting x — l-h cos 0, 

•» /2 - x\» 



A8)^f(^)'{(8-iy-^2xi8+l)-a^]dx (52), 



235 A.] CALCULATION OF KINETIC ENEBGY. 429 

we get 

T = \iraa^ [lf{8) (i,« + i/») + 2/(- s) W + ^Z^) 

+ 22[(««-cos»^)8m^d5(i,^. + i;£/)} (53). 

It will be seen that, while V in (39) is expressible by the 
squares only of the co-ordinates, a like assertion cannot in general 
be made of T, Hence J.,, £, &c. are iwt in general the normal 
co-ordinates. Nor could this have been expected. If, for example, 
we take the case where the spherical surface is bounded by two 
circles of latitude equidistant from the equator, symmetry shews 
that the normal co-ordinates are, not A and B^ but {A + B) and 
{A - B). In this case /(- 8) =/(«). 

A verification of (53) may readily be obtained in the particular 
case of « = 1, the surface under consideration being the entire 
sphere. Dropping the dashed letters, we get 

T = \yraa' [^ (i,» + A') + %AA] 

= i7r<ra*{ 4(i, + 4)« + f (i, - £,)«} (54). 

In this case the displacements are of the purely translatory and 
rotatory tjrpes already discussed, and the coiTectness of (54) may 
be confirmed. 

Whatever may be the position of the circles of latitude by 
which the surfieuse is bounded, the true t}rpes and periods of 
vibration are determined by the application of Lagrange's method 
to (39), (53). 

When one pole, e.g. ^ = 0, is included within the surface, the 
co-ordinates B vanish, and ^«, A^ become the normal co-ordinates. 
If we omit the dashed letters, the expression for T becomes 
simply 

T^\iraa''Lf{8)A,^ (55). 

From (43), (55) the frequencies of free vibrations for a hemi- 
sphere are immediately obtainable. The equation for A, is 

aa^f{8)A,'\-H{^-8){2^-l)A,^0 (56); 

so that, if Af vary as cos j),t, 

ir(^-^)j2^-l)^2nA» (^-^)(2^-l) , .. 



9 

430 CURVED PLATES OR SHELLS. [235 h. 

if we introduce the value of H from (40), and express a by means 
of the volume density p. 

In like manner for the saucer of 120°, from (44), 

^_ ^(^-^)(8^+4^^3) 
P' aa^f{8), 3*^^ ^^^)- 

The values of f(s) can be calculated without difficulty in the 
various cases. Thus, for the hemisphere, 

'2 

x-^ (4 - 4^ + ^ ) (1 4- 6a? - ^) da? 



/(2)=f 

.'1 



so that 



= 20 log 2 -12J = 1-52961, 
/(3) = 57^ - 80 log 2 = 1-88156, 
/(4) = 200 log 2 - 136J = 2-29609, &c. ; 



i).= -i^ X 5-2400, p3 = '^ X 14-726, p, = -^ x 28462. 

In experiment, it is the intervals between the various tones 
with which we are most concerned. We find 

j?,/p, = 2-8102, j3,/p, = 5-4316 (59). 

In the case of glass bells, such as are used with air-pumps, 
the interval between the two gravest tones is usually somewhat 
smaller ; the representative fraction being nearer to 2-5 than 2-8. 

For the saucer of 120°, the lower limit of the integral in (52) 
is f , and we get on calculation 

/(2) = -12864, /(3) = -054884, 
giving p, = J^ X 7-9947, p,= 'f— x 20'9U, 

P3:;>2 = 2-6157. 

The pitch of the two gravest tones is thus decidedly higher than 
for the hemisphere, and the interval between them is lees. 

With reference to the theory of tuning bells, it may be worth 
while to consider the effect of a small change in the angle, for the 
case of a nearly hemispherical bell In general 

4ir (jj» - Bf I sin-» tan** ^0 d0 

Ps' = p is ... (60). 

aV I tan" ^0 [(s + cos 0y + 2 sin« 0} sin 0d0 

• A 



235 A.] FREQUENCY EQUATION. 431 

If 5 = ^«ir + 8^, and P, denote the value of /), for the exact hemi- 
sphere, we get from previous results 



Thus 



;>.' = P^l 4- S^ If? -,:^g)] = p.' (1 -20 S^). 

shewing that an increase in the angle depresses the pitch. As to 
the interval between the two gravest tones, we get 



©' =(S/^^'^ •'''">• 



p. 

shewing that it increases with 0. This agrees with the results 
given above for = 60°. 

The fact that the form of the normal functions is independent 
of the distribution of density and thickness, provided that they 
vary only with latitude, allows us to calculate a great variety of 
cases, the difficulties being merely those of simple integration. If 
we suppose that only a narrow belt in co-latitude has sufficient 
thickness to contribute sensibly to the potential and kinetic 
energies, we have simply, instead of (60), 

P' a*a {(s + cos 0y + 2 sin* d] ^^ ^' 

whence ^ = 4,/l«±i^^^^:i^i (63). 

Pa V Ul+ecoS^-COS^^j ^ ^ 

The ratio varies very slowly from 3, when ^ = 0, to 2*954, when 

« 
If 2A denote the thickness at any co-latitude 0, Hoch\ aoc h, 
I have calculated the ratio of frequencies of the two gravest tones 
of a hemisphere on the suppositions (1) that A x cos d, and (2) that 
/i X (1 + cos 0), The formula used is that marked (60) with H and <r 
under the integral signs. In the first case, Pz:pi= 1*7942, diflFering 
greatly from the value for a uniform thickness. On the second 
more moderate supposition as to the law of thickness, 

p, : Pa = 2-4591, P4 : Pa = 4-4837. 



432 CUBVED PLATES OR SHELLS. [235 h. 

It would appear that the smallness of the interval between the 
gravest tones of common glass bells is due in great measure to the 
thickness diminishing with increasing 0, 

It is worthy of notice that the curvature of deformation 8 (p~0» 
which by (38) varies as sin"* ^ tan' J^, vanishes at the pole for 
9 = 3 and higher values, but is finite for « = 2. 

The present chapter has been derived very largely from 
various published memoirs by the author^ The methods have 
not escaped criticism, some of which, however, is obviated by 
the remark that the theory does not profess to be strictly 
applicable to shells of finite thickness, but only to the limiting 
case when the thickness is infinitely small. When the thickness 
increases, it may become necessary to take into account certain 
" local perturbations " which occur in the immediate neighbourhood 
of a boundary, and which are of such a nature as to involve 
extensions of the middle surface. The reader who wishes to 
pursue this rather difficult question may refer to memoirs by 
Love', Lamb', and Basset*. From the point of view of the present 
chapter the matter is perhaps not of great importance. For it 
seems clear that any extension that may occur must be limited to 
a region of infinitely small arec^ and affects neither the types nor 
the frequencies of vibration. The question of what precisely 
happens close to a free edge may require further elucidation, but 
this can hardly be expected fix)m a theory of thin shells. At 
points whose distance from the edge is of the same order as the 
thickness, the characteristic properties of thin shells are likely to 
disappear. 

^ Proe, Land, Math. Soe., ziii. p. 4, 1S81 ; xx. p. 872, 1889 ; Proe. Roy. Soe., vol. 
45, p. 105, 1888 ; vol. 45, p. 443, 1888. 

» Phil. Trans., 179(a), p. 491, 1888; Proe. Boy. Soc., vol. 49, p. 100, 1891; 
Theory of Elasticity , ch. xxi. 

< Proe. Lond. Math. Soc., voL xxi. p. 119, 1890. 

4 Phil. Trans. 181 (a), p. 433, 1890; Am. Math. Joum., vol xvi. p. 254, 1894. 



CHAPTER Xb. 



ELECTRICAL VIBRATIONS. 



236 i. The introduction of the telephone into practical use, 
and the numerous applications to scientific experiment of which 
it admits, bring the subject of alternating electric currents 
within the scope of Acoustics, and impose upon us the obligation 
of shewing how the general principles expounded in this work may 
best be brought to bear upon the problems presenting themselves. 
Indeed Electricity affords such excellent illustrations that the 
temptation to use some of them has already (^ 78, 92 a, 111 6) 
proved irresistible. It will be necessary, however, to take for 
granted a knowledge of elementary electrical theory, and to abstain 
for the most part from pursuing the subject in its application to 
vibrations of enormously high frequency, such as have in recent 
years acquired so much importance from the researches initiated 
by Lodge and by Hertz. In the writings of those physicists and in 
the works of Prof. J. J. Thomson^ and of Mr O. Heaviside' the 
reader will find the necessary information on that branch of the 
subject. 

The general idea of including electrical phenomena under those 
of ordinary mechanics is exemplified in the early writings of Lord 
Kelvin ; and in his " Dynamical Theory of the Electro-magnetic 
Field'" Maxwell gave a systematic exposition of the subject from 
this point of view. 

^ Recent Researches in Electricity and Magnetism^ 1893. 

2 Electrical Papers, 1892. 

3 Phil, Trans, vol. 155, p. 459, 1865 ; Collected Works, vol. 1, p. 526. 

K 28 



434 ELECTBICAL VIBRATIONS. [235^. 

236 J. We commence with the consideration of a simple 
electrical circuit, consisting of an electro-magnet whose terminals 
are connected with the poles of a condenser, or leyden^, of capacity 
C. The electro-magnet may be a simple coil of insulated wire, of 
resistance It, and of self-induction or inductance L, If there be an 
iron core, it is necessary to suppose that the metal is divided so as 
to avoid the interference of internal induced currents, and further 
that the whole change of magnetism is small'. Otherwise the 
behaviour of the iron is complicated with hysteresis, and its effect 
cannot be represented as a simple augmentation of Z. Also for 
the present we will ignore the hysteresis exhibited by many kinds 
of leydens. 

If X denote the charge of the leyden at time t, x is the 

current, and if EiCO^pt be the imposed electro-motive force, the 

equation is 

Lx-k-Rx-^xjC^^E^Qo&pt (1). 

The solution of (1) gives the theory oi forced electrical vibrations ; 
but we may commence with the consideration of the free vibra- 
tions corresponding to. JFi = 0. This problem has already been 
treated in § 45, from which it appears that the currents are 

oscillatory, if 

R<2^f(LIC) (2). 

The fiEtct that the discharges of leydens are often oscillatory was 
suspected by Henry and by v. Helmholtz, but the mathematical 
theory is due to Kelvin*. 

When R is much smaller than the critical value in (2), a large 

number of vibrations occur without much loss of amplitude, and 

the period t is given by 

T = 27rV(Ci) (3). 

In (2), (3) the data may be supposed to be expressed in c. G. s. 
electro-magnetic measure. If we introduce practical units, so 
that L', R\ C represent the inductance, resistance and capacity 
reckoned respectively in earth-quadrants or henrys, ohms, and 
microfarads*, we have in place of (2) 

iZ' < 2000 V(i7<?') (2'), 

1 This term has been approved by Lord Kelvin (*' On a New Form of Air Leyden 
A'C.*' Proc, Roy. Soc, vol. 62, p. 6, 1892). 

> Phil Mag,, vol. 23. p. 225, 1887. 

> " On Transient Electric Currents," Phil. Mag., June, 1868. 
* Ohm=10®, henry=10», microfarad =10"". 



235 y.] CALCULATION OF PERIOD. 435 

and in place of (3) 

T = 27r.l0-V(O'i') (3'). 

With ordinary appliances the value of t is very small ; but by 
including a considerable coil of insulated wire in the discharging 
circuit of a leyden composed of numerous glass plates Lodge ^ has 
succeeded in exhibiting oscillatory sparks of periods as long as 
T^ second. 

If the leyden be of infinite capacity or, what comes to the 
same thing, if it be short-circuited, the equation of free motion 
reduces to 

Lx + Rx^O (4); 

whence x = cho€r<^i^^* (5)», 

Xo representing the value of x when f = 0. The quantity L/R is 
sometimes called the time-constant of the circuit, being the time 
during which free currents fall oflF in the ratio of ^ : 1. 

Returning to equation (1), we see that the problem falls under 
the general head of vibrations of one degree of freedom, discussed 
in § 46. In the notation there adopted, w' = ((7i)~^ k^R/L, 
E^Ei/L] and the solution is expressed by equations (4) and (5). 
It is unnecessary to repeat at length the discussion already given, 
but it may be well to call attention to the case of resonance, 
where the natural pitch of the electrical vibrator coincides with 
that of the imposed force {p^LC=l). The first and third terms 
then (§ 46) compensate one another, and the equation reduces to 

Rx:=EiC03pt (6). 

In general, if the leyden be short-circuited (0= x ), 

El 
^ "^ L'p" + if » ^^ oospt+pL sin pt] (7); 

so that, if p much exceed R/L, the current is greatly reduced by 
self-induction. In such a case the introduction of a leyden of 
suitable capacity, by which the self-induction is compensated, 
results in a large augmentation of current*. The imposed electro- 
motive force may be obtained fix)m a coil forming part of the 
circuit and revolving in a magnetic field. 

1 Proc, Roy. Intt., March, 1889. 
^ Helmholtz, Pogg. Ann.j Lxxzni., p. 505, 1851. 

s Maxwell, ** Experiment in Magneto-Electric Indaction,'* Phil Mag,^ May, 
1868. 

28—2 



434 ELBCTRIUAL TIBRATIONa [235^. 

23S^'. We commenoe with the consideration of a simple 
electrical circuit, consisting of an electro-magnet whose terminals 
are connected with the poles of a condenser, or leaden', of capacity 
C. The electro'magnet may be a simple coil of insulated wire, of 
resistance R, and of self-induction or inductance L. If there be an 
iron core, it is necessary to suppose that the metal is divided so as 
to avoid the interference of internal induced currents, and further 
that the whole change of magnetism is small*. Otherwise the 
behaviour of the iron is complicated with hystereais, and its effect 
cannot be represented as a simple augmentation of L. Also for 
the present we will ignore the hysteresis exhibited by many kinds 
of leydens. 

If w denote the charge of the leyden at time t, x ia the 
current, and if EjCoapt be the imposed electro-motive force, the 
equation is 

Lx + R£ + x/C=E,coBpt (1). 

The solution of (1) gives the theory oi forced electrical vibrations; 
but we may commence with the consideration of the free vibra- 
tions corresponding to £, = 0. This problem has already been 
treated in § 43, ivom which it appears that the currents are 
osciUatory, if 

B<2V(i/C) (2). 

The fact that the discharges of leydens are often oscillatory was 
suspected by Henry and by v. Helmholtz, but the mathematical 
theory is due to Kelvin'. 

When R is much smaller than the critical value in (2), a lai^ 
number of vibrations occur without much loss of amplitude, and 
the period t is given by 

T~2v^(CL) (3). 

In (2), (3) the data may be supposed to be expressed in aQ.8. 
electro-magnetic measure. If we introduce practical units, so 
that L', R', C represent the inductance, resistance and capacity 
reckoned respectively in earth-quadrants or hesrys, ohms, ud 
microfarads*, we have in place of (2) 

R'<2000'J{L'IG') 

1 This term bu been approved by Lord KbItId (" On a h'e« Furm of Air 
4e." Proe. Roy. Soc., vol. 52, p. 6, 1892). 

• Phil. Mag., vol. 23, p. M5, 1887. 
> "On Irauiient Eleotric Cucrente," Phil, ilag^ Jvbm, 1858. 

* Ohm = I0*, henty = 10», microfarads 10"". 




235^.] CALCULATION OF PERIOD. 435 

and in place of (3) 

T = 27r.lO-'V(C'i;'> (3'). 

With ordinary appliances the value of t is very small ; but by 
iDCluding a considerable coil of insulated wire in the dischar^ng 
circuit of a leyden composed of numerous glass plates Lodge' has 
succeeded in exhibiting oscillatory sparks of periods as long as 
jj, second. 

If the leyden be of infinite capacity or, what comes to the 
same thing, if it be short-circuited, the equation of free motion 
reduces to 

LS + Ri~Q (4); 

whence i = jroe~**"'" (5)*, 

it representing the value of a: when f = 0. The quantity LfR is 
sometimes called the time-constant of the circuit, being the time 
during which free currents fell off in the ratio of e : 1. 

Returning to equation (1), we see that the problem falls under 
the general head of vibrations of one degree of freedom, discussed 
in § 46. In the notation there adopted, 7i' = (C7L)-', k^S/L, 
E^EijL; and the solution is expressed by equations (4) and (5). 
It is unnecessary to repeat at length the discussion already given, 
but it may be well to call attention to the case of resonance, 
where the natural pitch of the electrical vibrator coincides with 
that of the imposed force (p-LC—l). The first and third terms 
then <| 46) compensate one another, and the equation reduces to 
Ri = E,coapt (6). 

In general, if the leyden be short-circuited (0= oo ), 

A'j—rg.lRoospt+pLsinpt] (7); 

80 that, if p much exceed R/L, the current is greatly reduced by 
self-induction. In such a case the introduction of a leyden of 
suitable capacity, \iy which the self-induction is compensated, 
results in a large augmentation of eunent* The imposed electro- 
mol'ive force mM^m||^^^^mg^ forming part of the 
I circuit and revoS 



1 



436 ELECTBICAL VIBRATIONS. [235 J. 

In any circuit, where vibrations, whether forced or free, pro- 
portional to cos pt are in progress, we have a? = — ^a:, and thus the 
terms due to self-induction and to the leyden enter into the 
equation in the same manner. The law is more readily expressed 
if we use the stiffiiess /i, equal to 1/(7, rather than the capacity 
itself. We may say that a stiffness /i compensates an inductance 
Z, if fi^p^L, and that an additional inductance AZ is compensated 
by an additional stiffness A/i, provided the above proportionality 
hold good. This remark allows us to simplify our equations by 
omitting in the first instance the stiffness of leydens. When the 
solution has been obtained, we may at any time generalise it 
by the introduction, in place of Z, of Z — /fp"*, or Z — (p^CyK In 
following this course we must be prepared to admit negative 
values of Z. 

236 k. We will next suppose that there are two independent 
circuits with coefficients of self-induction Z, N, and of mutual 
induction Jlf , and examine what will be the effect in the second 
circuit of the instantaneous establishment and subsequent main- 
tenance of a current x in the first circuit. At the first moment 
the question is one of the function T only, where 

r = iZ^ + i¥iy + iiVy« (1); 

and by Kelvin's rule (§ 79) the solution is to be obtained by 
making (1) a minimum under the condition that x has the given 
value. Thus initially 

^ • /ox 

Vo^-^^ (2); 

and accordingly (§ 235 j) after time t 

y ^i^-^^/W (3), 

if S be the resistance of the circuit. The whole induced current, 
as measured by a ballistic galvanometer, is given by 



/ 



ydt=-^^ (4), 



S 

in which N does not appear. The current in the secondary circuit 
due to the cessation of a previously established steady current x in 
the primary circuit is the opposite of the above. 

A curious property of the initial induced current is at once 
evident from Kelvin's theorem, or from equation (2). It appears 



235 i.] SECONDARY CIRCUIT. 437 

that, if M be given, the initial current is gi-eatest when N is least. 
Further, if the secondary circuit consist mainly of a coil of n turns, 
the initial current increases with diminishing n. For, although 
Mxn, Nccn^\ and thus yoccl/n. In fact the small current 
flowing through the more numerous convolutions has the same 
effect as regards the energy of the iield as the larger current in the 
fewer convolutions. This peculiar dependence upon n cannot be 
investigated by the galvanometer, at least without commutators 
capable of separating one part of the induced current from the 
rest ; for, as we see from (4), the galvanometer reading is affected 
in the reverse direction. It is possible however to render evident 
the increased initial current due to a diminished n by observing 
the magnetizing effect upon steel needles. The magnetization 
depends mainly upon the initial maximum value of the current, 
and in a less degree, or scarcely at all, upon its subsequent 
duration, ^ 

The general equations for two detached circuits, influencing 

one another only by induction, may be obtained in the usual 

manner from (1) and 

F^iRx' + ^Sy' (5). 

Thus Lx + My + Rx^X) 

Mx-¥Ny + Sy=Y] ^^^• 

These equations, in a more general form, are considered in 
§ 116. If a harmonic force X = e^P^ act in the first circuit, and 
the second circuit be free from imposed force (F = 0), we have on 
elimination of y 

shewing that the reaction of the secondary circuit upon the first is 
to reduce the inductance by 

P'p^N^'+S^ ^^^'^ 

and to increase the resistance by 

P p'N' + S' ^^^• 

* PhU, Mag., vol. 38, p. 1, 1869; vol. 39, p. 428, 1870. 

' Maxwell, Phil. Trant., voL 165, p. 459, 1865, where, however, J/« is mis- 
printed M. 



438 ELECTRICAL VIBRATIONS. [235 i. 

The formulae (8) and (9) may be applied to deal with a more 
general problem of considerable interest, which arises when (as in 
some of Henry's experiments) the secondary circuit acts upon a 
third, this upon a fourth, and so on, the only condition being that 
there must be no mutual induction except between immediate 
neighbours in the series. For the sake of distinctness we will 
limit ourselves to four circuits. 

In the fourth circuit the current is due ex hypothesi only to 
induction from the third. Its reaction upon the third, for the rate 
of vibration under contemplation, is given at once by (8) and (9) ; 
and if we Use the complete values applicable to the third circuit 
under these conditions, we may thenceforth ignore the foiurth 
circuit. In like manner we can now deduce the reaction upon 
the secondary, giving the effective resistance and inductance of 
that circuit under the influence of the third and fourth circuits ; 
and then, by another step of the same kind, we may arrive at the 
values applicable to the primary circuit, under the influence of all 
the others. The process is evidently general ; and we know by 
the theorem of § 111 6 that, however extended the train of circuits, 
the influence of the others upon the first must be to increase its 
effective resistance and diminish its effective inertia, in greater 
and greater degree as the frequency of vibration increcises. 

In the limit, when the frequency increases indefinitely, the 
distribution of currents is determined by the induction-coefficients, 
irrespective of resistance, and, as we shall see presently, it is of 
such a character that the currents are alternately opposite in sign 
as we pass along the series. 

235 L Whatever may be the number of independent currents, 
or degrees of freedom, the general equations are always of the 
kind already discussed §§ 82, 103, 104, viz. 

d dT dF dV ^ y .-. 

dtdx di dx " ^ ^' 

where T, F, Fare (§ 82) homogeneous quadratic functions. In (1) 
the co-ordinates x^, x^, ... denote the whole quantity of electricity 
which has passed at time t, the currents being Xi, x^, &c. When 
F=0, it is simpler to express the phenomena by means of the 
currents. Thus, in the problem of steady electric flow where all 



235 Z.] INITIAL CURRENTS. 439 

the quantities Xy representing electro-motive forces, are constant, 
the currents are determined directly by the linear equations 

dF/ddk^X,, dFldx,^X^,Sic (2). 

On the other hand when the question under consideration is 
one of initial impulsive effects, or of forced vibrations of ex- 
ceedingly high frequency, everything depends upon T, and the 
equations reduce to 

d djL -^ d dT ** p ^^ivv 

As an example we may consider the problem, touched upon at 
the close of § 235 k, of a train of circuits where the mutual induc- 
tion is confined to immediate neighbours, so that 

r= ^ix^ -I- iaaa?a* + \an^% + • • • 

coefficients such as ai„ 014, o^ not appearing. If o^ be given, 

either as a current suddenly developed and afterwards maintained 

constant, or as a harmonic time function of high frequency, while 

no external forces operate in the other circuits, the problem 

is to determine x^, x^, &c. so as to make T as small as possible, 

§ 79. The equations are easily written down, but the conclusion 

aimed at is perhaps arrived at more instnictively by consideration 

of the function T itself. For, T being homogeneous in Xi,X2, &c., 

we have identically 

rtm dT dT ,»v 

2r=..3^+x,^+ ('')• 

And, since when T is a minimum, dT/dx^, dT/dx^y &c, all vanish, 

But if a?,, x^, &C., had all been zero, 2T would have been equal to 
OiiXi^ It is clear therefore that ai^XiX^ is negative ; or, as a,, is 
taken positive, the sign of x^ is the opposite to that of Xi, 

Again supposing Xi, x^ both given, we must, when T is a 
minimum, have dT/dx^, dT/dx^, &c., equal to zero, and thus 

Z^min. = chi^' + 2auXiX^ + d^x^ + 2a«a:2«8. 
As before, 2,T might have been 

OiiO?!* -I- iOr^^X^X^ + CLnX^, 
> The dots are omitted as mmecesaary. 



440 ELECTRICAL VIBRATIONS. [235 L 

simply. The minimum value is necessarily less than this, and 
accordingly the signs of x^ and a?, are opposite. This argument 
may be continued, and it shews that, however long the series may 
be, the induced currents are alternately opposite in sign^ a result 
in harmony with the magnetizations observed by Henry. 

In certain cases the minimum value of T may be very nearly 
zero. This happens when the coils which exercise a mutual 
inductive influence are so close throughout their entire lengths 
that they can produce approximately opposite magnetic forces at 
all points of space. Suppose, for example, that there are two 
similar coils A and -B, each wound with a double wire (Ai, -4$), 
(Bi, -Bo), and combined so that the primary circuit consists of -4i, 
the secondary of A^ and Bi joined by inductionless leads, and the 
tertiary of -Bj simply closed upon itself. It is evident that T is 
made approximately zero by taking a?2 = — ir^ and 0?, = — a-jsa?!. 
The argument may be extended to a train of such coils, however 
long, and also to cases where the number of convolutions in 
mutually reacting coils is not the same. 

In a large class of problems, where leyden effects do not occur 
sensibly, the course of events is determined by T and F simply. 
These functions may then be reduced to sums of squares ; and the 
typical equation takes the form 

ax + bi=^X (6). 

If X = 0, that is if there be no imposed electro-motive forces, the 

solution is 

i^d^^e-^ia (7) 

Thus any system of initial currents flowing whether in detached 
or connected linear conductors, or in solid conducting masses, may 
be resolved into " normal " components, each of which dies down 
exponentially at its own proper rate. 

A general property of the "persistences," equal to a/6, is 
proved in § 92 a. For example, any increase in permeability, due 
to the introd\iction of iron (regarded as non-conducting), or any 
diminution of resistance, however local, will in general bring about 
a rise in the values of all the persistences'. 

In view of the discussions of Chapter v. it is not necessary to 
dwell upon the solution of equations (1) when X is retained. The 

1 Phil Mag., vol. 38, p. 18, 1869. 

2 Brit. Astoc. Report, 1885, p. 911. 



235 Z.] TWO CONDUCTORS IN PARALLEL. 441 

reciprocal theorem of § 109 has many interesting electrical appli- 
cations ; but, after what has there been said, their deduction will 
present no difficulty. 

236 7/1. In § 111 b one application of the general formulae to 
an electrical system has already been given. As another example, 
also relating to the case of two degrees of freedom, we may take 
the problem of two conductors iii jmrallel. It is not necessary to 
include the influence of the leads outside the points of bifurcation; 
for provided that there be no mutual induction between these parts 
and the remainder, their inductance and resistance enter into the 
result by simple addition. 

Under the sole operation of resistance, the total current a.\ 

would divide itself between the two conductors (of resistances R 

and S)ivL the parts 

S , R 

R + S""' ^"""^ R + S""'' 

and we may conveniently so choose the second co-ordinate that 
the currents in the two conductors are in general 

^-^-;y^i + ^, and ^-^^i-^, 

a?i still representing the total current in the leads. The dissi- 
pation-function, found by multiplying the squares of the above 
currents by ^R, ^S respectively, is 

Da 

F = ^j^_^^x,^ + i(R + S)x,' ^^>- 

Also, L, M, N being the induction coefficients of the two 
branches, 

■'"* {R+sy '''' 

Thus, in the notation of § 111 6, 

_ LS^^MRS + NR' _ (L^M)S + (M^N)R 

^^" {R-^-Sy ~' ^"" R + 8 

ill = ji^g* ^13 = 0, 6a = 12 + 5. 



442 ELECTRICAL VIBRATIONS. [235 m. 

Accordingly by (5), (8) § 111 6, 

^,_L S' + 2MRS + NR' {{L^M)S + (M-'N)R\* 
(R + Sy {R + Sy{L^2M + N) 

{(L^M)8-^(M^N)R]^ 
^(L^2M-\'N)[{R + Sy+i^(L-2M+Ny}"'^ ^' 

These are respectively the effective resistance and the effective 
inductance of the combination ^ It is to be remarked that 
(L — 2M+N') is necessarily positive, representing twice the kinetic 
energy of the system when the currents in the two conductors 
are + 1 and — 1. 

The expressions for R and L' may be put into a form* which 
for many purposes is more convenient, by combining the com- 
ponent fractional terms. Thus 

J., _ RS( R + S)-^p^{R(M-'Ny + S(L^My} , 

(R + Sy-^p'{L-^2M-¥Ny ^ ^' 

(R+sy+p'(L-2M+Ny •••^ ^' 

in which (LN — M^) is positive by virtue of the nature of T. 

As p increases from zero, we know by the general theorem 
§ 111 6, or from the particular expressions (3), (4), that Rf con- 
tinually increases and that L' continually decreases. 

When p is very small, 

p, RS ,, LS' + 2MRS±NR^ ... 

. ^^rTs' ^^ W+sy ^^^- 

In this case the distribution of the main current between the 
conductors is determined by the resistances, and (§ 111 b) the values 
of R and L' coincide respectively with 2F/xi\ 2T/xi\ The resist- 
ance is manifestly the same as if the currents were steady. 

On the other hand, when p is very great, 

^, R(M^Ny-^8(L-My LN^M^ 

In this case the distribution of currents is independent of the 
resistances, being determined in accordance with Kelvin's theorem 

1 PhiL Mag,, vol. 21. p. 877, 18S6. 
> J. J. Thomson, loc. cit. § 421. 



235 m.] CONTIGUOUS wires. 443 

in such a manner that the ratio of the currents in the two con- 
ductors is (J\r— -10 ' (^ "" ^)' ^ when p is small, the values in 
(6) coincide with 2Flx,\ ^.Tjx^K 

When the two wires composing the conductors in parallel are 
wound closely together, the energy of the field under high fre- 
quency may be very small. There is an interesting distinction to 
be noted here dependent upon the manner in which the con- 
nections are made. Consider, for example, the case of a bundle 
of five contiguous wires wound into a coil, of which three wires, 
connected in series so as to have maximum inductance, constitute 
one of the branches in parallel, and the other two, connected 
similarly in series, constitute the other branch. There is still an 
alternative as to the manner of connection of the two branches. 
If steady currents would circulate opposite ways (if negative), the 
total current is divided into two parts in the ratio 3 : 2, in such a 
manner that the more powerful current in the double wire nearly 
neutralises at external points the magnetic effects of the less 
powerful current in the triple wire, and the total energy of the 
system is very small. But now suppose that the connections are 
such that steady currents would circulate the same way in both 
branches {M positive). It is evident that the condition of mini- 
mum energy cannot be satisfied when the currents are in the same 
direction, but requires that the smaller current in the triple wire 
should be in the opposite direction to that of the larger current in 
the double wire. In fact the currents must be as 3 to — 2 ; so 
that (since on the same scale the total current is unity) the 
component currents in the branches are both numerically greater 
than the total current which is algebraically divided between 
them. And this peculiar feature becomes more and more strongly 
marked the nearer L and N approach to equality^ 

The unusual development of currents in the branches is, of 
course, attended by an augmented effective resistance. In the 
limiting case when the m convolutions of one branch are supposed 
to coincide geometrically with one another and with the n convo- 
lutions of the second branch, we have 

and from (6) R='^'^-^^ (7). 

(m — n)* 

1 PhiL Mag., vol. 21, p. 876, 1SS6. 



444 ELECTBICAL VIBRATIONS. [235 m. 

an expression which increases without limit, as m and n approach 
to equality. 

The fact that under certain conditions the currents in both 
branches of a divided circuit may exceed the current in the mains 
has been verified by direct experiments Each of the three 
currents to be compared traversed short lengths of similar German- 
silver wire, and the test consisted in finding what lengths of this 
wire it was necessaiy in the various cases to include between the 
terminals of a high resistance telephone in order to obtain sounds 
of equal intensity. The variable currents were derived from a 
battery and scraping contact apparatus (§ 235 r), directly included 
in the main circuit. 

The general formulae (3'), (4') undergo simplification when the 
conductors in parallel exercise no niutual induction. Thus, when 

_ RS(R + S)+ p'(RN' + S D) 






(9). 



If further iV^ = 0, (8) and (9) reduce to 

S[R{R^8) + l^D] _ LS^ 

(R + Sy-^-p'L' / (R + Sy-^p'L^'"^^^^' 

The peculiar features of the combination are brought out most 
strongly when S, the resistance of the inductionless component, is 
great in comparison with R. In that case if the current be steady 
or slowly vibrating, it flows mainly through R, while the resistance 
and inductance of the combination approximate to R and L respec- 
tively ; but if on the other hand the current be a rapidly vibrating 
one, it flows mainly through S, so that the resistance of the combi- 
nation approximates to S, and the inductance to zero. These 
conclusions are in agi'eement with (10). 

If the branches in parallel be simple electro-magnets, L and N 
are necessarily positive, and the numerator in (9) is incapable of 
vanishing. But, as we have seen, when leydens are admitted, this 
restriction may be removed. An interesting case arises when the 
second branch is inductionless, and is interrupted by a leyden of 

1 Phil, Mag., vol. 22, p. 495, 1886. 



235 m.] CONDUCTORS in parallel. 445 

capacity (7, so that i\r= — (Cj»^)-\ while at the same time R^S. 
The latter condition reduces the numerator in (9) to 

Thus L' vanishes, (i) when LCp^ = 1, and (ii) when CR^ = L. The 
first alternative is the condition that the loop circuit, considered 
by itself, should be isochronous with the imposed vibrations. 
The second expresses the equality of the time-constants of the two 
branches. If they be equal, the combination behaves like a simple 
resistance, whatever be the character of the imposed electro- 
motive forced 

235 n. When there are more than two conductors in parallel, 
the general expressions for the equivalent resistance and induc- 
tance of the combination would be very complicated ; but a few 
particular cases are worthy of notice. 

The first of these occurs when there is no mutual induction 
between the members. If the quantities relating to the various 
branches be distinguished by the suffixes 1, 2, 3, ..., and if E be 
the difference of potentials at the common terminals, we have 

E = (ipLi + Ri) Xi = (ipL^ + K) X2= (1); 

by which R and L' are determined. Thus, if we write 

^ R' + p'D ' ^ R^-^-fD ^'^^^ 

we have from (2) 

,' A , B 
^'" A^^W' ^'"A' + P^B' ^*^- 

Equations (3) and (4) contain the solution of the problem^ 
When p = 0, 

When on the other hand p is very great, 

1{RL-^ 1 
^'{t(L-")Y' "^{L-') ^ ^* 

^ Chrystal, **0n the Differential Telephone," FMn. Tram,, vol. 29, p. 616, 
ISSO. 

> Phil, Mag., vol. 21, p. 879, 18S6. 



448 ELECTRICAL VIBRATIONS. [235 0. 

From this it appears that a want of balance depending on Oi^ 
cannot compensate for the action of the third circuit, so as to 
produce silence in the secondary circuit, unless 6, be negligible 
in comparison with pa^, that is unless the time-constant of the 
third circuit be very great in comparison with the period of the 
vibration. Otherwise the effects are in different phases, and 
therefore incapable of balancing. 

We will now introduce a fourth circuit, and suppose that the 
primary and secondary circuits are accurately conjugate, so that 
Oia = 0, and also that the mutual induction 0,4 between the third 
and fourth circuits may be neglected. Then 

ip {a^x.i + a^x^ + h^x^ == — ipa^zXi, 

ip {a4sX.2 + a^x^) + b^x^ = — ipOi^Xi ; 
whence 



p'a,^^ 






ipau-\-hS 



It appears that two conditions must be satisfied in order to 
secure a balance, since both the phases and the intensities of the 
separate effects must be the same. The first condition requires 
that the time-constants of the third and fourth circuits be equal, 
unless indeed both be very great, or both be very small, in com- 
parison with the period. If this condition be satisfied, balance 
ensues when 

^i.^ + ^?^-»* = (4); 

and it is especially to be noted that the adjustment is independent 
of pitch, so that (by Fourier's theorem) it suffices whatever be the 
nature of the variable currents operative in the primary. 

As regards the position of the third and fourth circuits, usually 
represented by coins in illustrative experiments, it will be seen 
from the symmetry of the right-hand member of (3) that the 
middle position between the primary and secondary coils is suit- 
able, inasmuch as the product a^^a^^ is stationary in value when 
the coin is moved slightly so as to be nearer say to the primary 




235 0.] GENERALIZED RESISTANCE. 449 

and further from the secondary ^ Approximate independence of 
other displacements is secured by the geometrical symmetry of the 
coils round the axis. 

235 p. For the accurate comparison of electrical quantities 
the "bridge" arrangement of Wheatstone is usually the most 
convenient, and is equally available with the galvanometer in the 
case of steady or transitory currents, or with the telephone in the 
case of periodic currents. Similar effects may be obtained in most 
cases without a bridge by the employment of the differential 
galvanometer or the differential telephone*. 

In the ordinary use of the bridge the four members a, b, c, d 
combined in a quadrilateral Fig. (53 a) are 
simple resistances. The battery branch/ l^ ^ 

joins one pair of opposite comers, and the 
indicating instrument is in the "bridge" 
e joining the other pair. " Balance " is 
obtained, when ad = be. But for our 
purpose we have to suppose that any 
member, e.g. a, is not merely a resistance, 
or even a combination of resistances. It may include an electro- 
magnet, and it may be interrupted by a leyden. But in any case, 
so long AS the current x is strictly harmonic, proportional to e**^*, 
the general relation between it and the difference of potentials V 
at the extremities is given by 

V=(ai'\-ia^)x (1), 

where a, and ia, are the real and imaginary parts of a complex 
coefficient a, and are functions of the frequency pl27r. In the 
particular case of a simple conductor, endowed with inductance L, 
Oi represents the resistance, and a^ is equal to pL. In general, ctj 
is positive; but a^ may be either positive, as in the above ex- 
ample, or negative. The latter case arises when a resistance R is 
interrupted by a leyden of capacity C, Here ai = -B, Oj = — l/pC, 
If there be also inductance Z, 

Qi^R, a^^pL-l/pC (2). 

As we have already seen, § 235 j, a^ may vanish for a particular 
frequency, and the combination is then equivalent to a simple 

1 See Lodge, Phil Mag,, vol. 9, p. 123, 18S0. 
' Chrystal, Edin. Traru,, loc, cit. 

R. y 29 



^ 



450 ELECTRICAL VIBRATIONS. [235 p. 

resistance. But a variation of frequency gives rise to a positive 
or negative aj. 

In all electrical problems, where there is no mutual induction, 

the generalized quantities, a, 6, &c., combine, just as they do when 

they represent simple resistances^ Thus, if a, a be two complex 

quantities representing two conductors in series, the corresponding 

quantity for the combination is {a + a). Again, if a, a represent 

two conductors in parallel, the reciprocal of the resultant is given 

by addition of the reciprocals of a, a. For, if the currents be a?, x\ 

corresponding to a difference of potentials V at the common 

terminals, 

V^ax — ax\ 

so that x-^x ^ ^(1/^ + V^')- 

In the application to Wheatstone's combination of the general 
theory of forced vibrations, we will limit the impressed forces to 
the battery and the telephone branches. If x, y be the currents 
in these branches, X, Y the corresponding electro-motive forces, 
we have, § 107, linear relations between Xy y, and X, F, which may 
be written 



X^Ax-¥By 
Y=Bx + Cy 



I (3), 



the coeflBcient of y in the first equation being identical with that 
of X in the second equation, by the reciprocal property. The three 
constants A,B^C are in general complex quantities, functions of/). 

The reciprocal relation may be interpreted as follows. If 
7 = 0, 5a: + (7y = 0, and 

y^B^^Ac ^^>- 

In like manner, if we had supposed X = 0, we should have 
found 

""'W^TaC ^^^' 

shewing that the ratio of the current in one member to the electro- 
motive force operative in the other is independent of the way in 
which the parts are assigned to the two members. 

1 For a more complete discussion of this subject see Heaviside ** On Resistance 
and Conductance Operators," PhiL Mag,, vol 24, p. 479, 1887 ; Electrical Papers, 
vol. n., p. 355. 



V 

\ 



235 p.] wheatstone's bridge. 451 

We have now to determine the constants A, J5, C in terms of 
the electrical properties of the system. If y be maintained zero 
by a suitable force F, the relation between x and X is X = Ax. 
A therefore denotes the (generalized) resistance to any electro- 
motive force in the battery member, when the telephone member is 
open. This resistance is made up of /, the resistance in the 
battery member, and of that of the conductors a-^c^ 6 + d, 
combined in parallel. Thus 

^=/+(«_±«HL+_^) (6). 

In like manner 

^^^^ (a.f6)(c4-d) 

a-^-o-hc + d ^ ' 

To determine B let us consider the force Y which must act 
in e in order that the current through it may be zero, in spite 
of the operation of X. We have Y^Bx, The total current x 
flows partly along the branch (a -he), and partly along (6-l-(i). 
The current through (a -he) is 

x\{a -f c) {b'^d)x_ 
l/(a-hc) + l/(6-hd) a + 6-hc + d ^ ^' 

and that through (6 + d) is 

The difference of potentials at the terminals of «, supposed to 
be interrupted, is thus 

c (6 + d) a; — d (a -h c) a? 
a+6+c+d ' 

and accordingly £=_J|^^ (10). 

By (6), (7), (10) the relationship of X, Fto a?, y is completely 
determined. 

The problem of the bridge requires the determination of the 
current y as proportional to X, when F=0, that is when no 
electro-motive force acts in the bridge itself; and the solution is 
given at once by the introduction into (4) of the values of -4, 5, C 
from (6), (7), (10). 

If there be an approximate " balance," the expression simplifies. 
For (6c — ad) is then small, and 5* may be neglected relatively to 

29—2 




452 ELECTRICAL VIBRATIONS. [235 2>. 

AC in the denominator of (4). Thus, as a sufficient approximation 
in this case, we may write 

y.^A (1^) an 

X"" AC" (6)x(7) ^ ^• 

The following interpretation of the process leads very simply 
to the approximate form (11), and is independent of the general 
theory. Let us first inquire what electro-motive force is necessary 
in the telephone member to stop any current through it. If such 
a force act, the conditions are, externally, the same as if the 
member were open ; and the current x in the battery member due 
to a force equal to X in that member is X/A, where A is written 
for brevity as representing the right-hand member of (6). The 
difference of potentials at the terminals of e, still supposed to be 
open, is found at once when x is known. It is given by 

cx(8)«dx(9) = 5ar, 

where B is defined by (10). In terms of X the difference of 
potentials is thus BXjA. If e be now closed, the same fraction 
expresses the force necessary in e in order to prevent the genera- 
tion of a current in that member. 

The case with which we have to deal is when X acts in /and 
there is no force in e. We are at liberty, however, to suppose that 
two opposite forces, each of magnitude BX/A, act in e. One of 
these, as we have seen, acting in conjunction with X in/, gives no 
current in e ; so that, since electro-motive forces act independently 
of one another, the actual cuirent in e, closed without internal 
electro-motive force, is simply that due to the other component. 
The question is thus reduced to the determination of the current 
in e due to a given force in that member. 

So far the argument is rigorous ; but we will now suppose that 
we have to deal with an approximate balance. In this case a force 
in e gives rise to very little current in / and in calculating the 
current in e, we may suppose /to be broken. The total resistance 
to the force in e is then given simply by C of equation (7), and the 
approximate value for y is derived by dividing — BX/A by C, as 
we found in (11). 

A continued application of the foregoing process gives y/X in 
the form of an infinite geometric series : — 



^ = - 



B 



, B' B' ) B 



Z" AC\ '^ AC A'C'^ '"] " B'-'AC' 



\ 



%. 
\ 



235 p.] APPROXIMATE BALANCE. 453 

This is the rigorous solution already found in (4) ; but the first 
term of the series suffices for practical purposes. 

The form of (11) enables us at once to compare the effects of 
increments of resistance and of inductance in disturbing a balance. 
For let ad = be, and then change d to d + d\ where d' = di' + id^. 
The value of y/X Ls propoitional to d', and the amplitude of the 
vibratory current in the bridge is proportional to mod. d\ that is, 
bo VW^ + da'^. Thus di', dj' are equally efficacious when nu- 
merically equals In most cases where a telephone is employed, 
the balance is more sensitive to changes of inductance than to 
changes of resistance. 

In the use of the Wheatstone balance for purposes of measure- 
ment, it is best to make a equal to c. The equality of b and d can 
then be tested by interchange of a and c, independently of the 
exactitude of the equality of these quantities. Another advantage 
lies in the fact that balance is independent of mutual induction 
between a and c or between b and d. 

235 q. In the formulae of § 235 p it has been assumed that 
there is no mutual induction between the various members of the 
combination. The more general theory has been considered very 
fully by Heaviside', but to enter upon it would lead us too far. 
It may be well, however, to sketch the theory of the arrangement 
adopted by Hughes, which possesses certain advantages in dealing 
with the electrical properties of wires in short lengths'. 

The apparatus consists of a Wheatstone's quadrilateral, Fig. 53 6, 
with a telephone in the bridge, one of the 
sides of the quadrilateral being the wire ^^^* ^^ ^ 

or coil under examination (P), and the 
other three being the parts into which a 
single German-silver wire is divided by 
two sliding contacts. If the battery- 
branch (B) be closed, and a suitable in- 
terrupter be introduced into the telephone- 
branch (T), balance may be obtained by 
shifting the contacts. Provided that the 
interrupter introduces no electro-motive 

^ '*0n the Bridge Method in its Application to Periodic Electric Carrents.*' 
Proc, Roy, Soc, vol. 49, p. 208, 1891. 

2 ''On the Self-induction of Wires," Part VI.; P/it7. Jf a^. , Feb. 1887; Electrical 
Papen, 1892, vol. ii., p. 281. 

' Journ. Tel, Eng,, vol. xv. (1886) p. 1 ; Proc, Roy, Soc, vol. xl. (1886) p. 451. 




f\ 



454 ELECTRICAL VIBRATIONS. [235 q, 

force of its own^, the balance indicates the proportionality of 

the four resistances. If P be the unknown resistance of the 

conductor under test, Q, R the resistances of the adjacent parts of 

the divided wire, S that of the opposite part (between the sliding 

contacts), then, by the ordinary rule, PS= QR; while Q, R, S are 

subject to the relation 

Q-^R + S^W, 

W being a constant. If now the interrupter be transferred from 
the telephone to the battery-branch, the balance is usually dis- 
turbed on account of induction, and cannot be restored by any 
mere shifting of the contacts. In order to compensate the 
induction, another influence of the same kind must be intro- 
duced. It is here that the peculiarity of the apparatus lies. A 
coil (not shewn in the figure) is inserted in the battery and another 
in the telephone-branch which act inductively upon one another, 
and are so mounted that the effect may be readily varied. The 
two coils may be concentric and relatively movable about the 
common diameter. In this case the action vanishes when the 
planes are perpendicular. If one coil be verj' much smaller than 
the other, the coefficient of mutual induction M is proportional to 
the cosine of the angle between the planes. By means of the 
two adjustments, the sliding of the contacts and the rotation of the 
coil, it is usually possible to obtain a fair silence. 

Hughes interpreted his observations on the basis of an as- 
sumption that the inductance of P was represented by M, irre- 
spective of resistance, and that the resistance to variable currents 
could (as in the case of steady currents) be equated to QR/S. 
But the matter is not quite so simple. The true formulae are, 
however, readily obtained for the case where the only sensible 
induction among the sides of the quadrilateral is the inductance L 
of the conductor P, 

Since there is no current through the bridge, there must be 
the same current (x) in P and in one of the adjacent sides (say) iJ, 
and for a like reason the same cunent y in Q and S, The differ- 
ence of potentials at time t between the junction of P and R and 
the junction of Q and S may be expressed by each of the three 
following equated quantities : — 

^ A condition not always satisfied in practice. 



i 



2355.] hughes' arrangement. 455 

Introducing the assumption that all the quantities vary har- 
monically with frequency p/27r, and eliminating the ratio y : a?, we 
find as the conditions required for silence in the telephone 

QR^SP = fML (1), 

M(P + Q + R'^S)^SL (2). 

It will be seen that the ordinary resistance balance (SP = QR) 
is departed from. The change here considered is peculiar to the 
apparatus and, so far as its influence is concerned, it does not 
indicate a real alteration of resistance in the wire. Moreover, 
since p is involved, the disturbance depends upon the rapidity of 
vibration, so that in the case of ordinary mixed sounds silence can 
be attained only approximately. Again, from the second equation 
we see that M is not in general a correct measure of the value 
oiL\ 

If, however, P be known, the application of (2) presents no 
diflSculty. In many cases we may be sure beforehand that P, 
viz. the effective resistance of the conductor, or combination of 
conductors, to the variable currents, is the same as if the currents 
were steady, and then P may be regarded as known. But there 
are other cases, — some of them will be alluded to below — in 
which this assumption cannot be made; and it is impossible to 
determine the unknown quantities L and P from (2) alone. We 
may then fall back upon (1). By means of the two equations 
P and L can always be found in terms of the other quantities. 
But among these is included the frequency of vibration ; so that 
the method is practically applicable only when the interrupter is 
such as to give an absolute periodicity. A scraping contact, 
otherwise very convenient, is thus excluded; and this is un- 
doubtedly an objection to the method. 

If the member P be without inductance, but be interrupted by 
a leyden of capacity C, the same formulae may be employed, with 
substitution of — 1/p'C for L. Equation (1) then gives a measure 
of C which is independent of the frequency. 

236 r. The success of experiments with this kind of apparatus 
depends very largely upon the action of the interrupter by which 
the currents are rendered variable. When periodicity is not 

* "Disoassion on Prof. Hughes' Address." Journ. TeL Eng., vol. xv., p. 54, 
Feb., 18S6. 



456 ELECTRICAL VIBRATIONS. [235 r. 

necessary, a scraping contact, actuated by a clock or by a small 
motor, answers very well; but it is advisable, following Lodge 
and Hughes, so to arrange matters that the current is suspended 
altogether at short intervals. The faint scraping sound, heard in 
the neighbourhood of a balance, is more certainly identified when 
thus rendered intermittent. 

But for many of the most interesting experiments a scraping 
contact is unsuitable. When the inductance and resistance under 
observation are rapidly varying functions of the frequency, it is 
evident that no sharp results are possible without an interrupter 
giving a perfectly regular electrical vibration. With proper appli- 
ances an absolute silence, or at least one disturbed only by a slight 
sensation of the octave of the principal tone, can be arrived at 
under circumstances where a scraping contact would admit of no 
approach to a balance at all. 

Tuning-forks, driven electromagnetically with liquid or solid 
contacts (§ 64), answer well so long as the frequency required 
does not exceed (say) 300 per second ; but for experiments with the 
telephone we desire frequencies of from 500 to 2000 per second. 
Good results may be obtained with harmonium reed interrupters, 
the vibrating tongue making contact once during each period 
with a stop, which can be adjusted exactly to the required position 
by means of a screws 

But perhaps the best interrupter for use with the telephone is 
obtained by taking advantage of the instability of a jet of fluid. 
If the diameter and the speed be chosen suitably, the jet may be 
caused to resolve itself into drops under the action of a tuning- 
fork in a perfectly regular manner, one drop corresponding to 
each complete vibration of the fork. Each drop, as it passes, 
may be made to complete an electric circuit by squeezing itself 
between the extremities of two fine platinum wires. If the 
electro-motive force of the battery be pretty high, and if the 
jet be salted to improve its conductivity, suflScient current passes, 
especially if the aid of a small step-down transformer be invoked. 
Finally the apparatus is made self-acting by bringing the fork 
under the influence of an electro-magnet, itself traversed by the 
same intermittent current. Such an apparatus may be made to 
work with frequencies up to 2000 per second, and it possesses 
many advantages, among which may be mentioned almost icbsolute 

1 Phil. Mag,, vol. 22, p. 472, 1SS6. 



235 r.] INTERRUPTEBS. 457 

constancy of pitch, and the avoidance of loud aerial disturbance. 
The principles upon which the action of this, interrupter depends 
will be further considered in a subsequent chapter. 

236^. Scarcely less important than the interrupter are the 
arrangements for measuring induction, whether mutual induc- 
tion, as required in § 235 g, or self-induction. Inductometers, as 
Heaviside calls them, may be conveniently constructed upon 
the pattern of Hughes. A small coil is mounted so that one 
diameter coincides with a diameter of a larger coil, and is 
movable about that diameter. The mutual induction M between 
the two circuits depends upon the position given to the smaller 
coil, which is read by a pointer attached to it, and moving over a 
graduated circle. If the smaller coil were supposed to be infinitely 
small, the value of if, as has already been stated, would be pro- 
portional to the sine of the displacement from the zero position 
(if =sO). But an approximation to this state of things is not 
desirable. If the mean radius of the small coil be increased until 
it amounts to '55 of that of the larger, not only is the efiBciency 
much enhanced, but the scale of M is brought to approximate 
coincidence, over almost the whole practical range, with the scale 
of degrees^ The absolute value of each degree may be arrived at 
in various ways, perhaps most simply by adjusting the mutual 
induction of the instrument to balance a standard of mutual 
induction 

For experiments upon the plan of § 235 q the one coil is 
included in the telephone and the other in the battery branch, 
but when the object is to secure a variable and measurable 
inductance, the two coils are connected in series. The inductance 
of the combination is then X + 2if+JV, of which the first and 
third terms are independent of the relative position of the coils. 

236 t Good results by the method of § 235 q have been 
obtained by Weber", and by the author' using a reed interrupter 
of frequency 1050 per second ; but the fact that inductance and 
resistance are mixed up in the measurements is a decided draw- 
back, if it be only because the readings require for their interpre- 
tation calculations not readily made upon the spot. 

1 Phil. Mag., vol. 22, p. 498, 1886. 

3 Electrical Review, April 9, Jalj 9, 1886. 

' PhiU Mag., loe. ciU 



458 ELECTRICAL VIBRATIONS. [235 t 

The more obvious arrangement is one in which both the 
induction and the resistance of the branch containing the subject 
under examination are in every case brought up to the given 
totals necessary for a balance. To carry this out conveniently we 
require to be able to add inductance without altering resistance, 
and resistance without altering inductance, and both in a measur- 
able degree. The first demand is easily met. If we include in 
the circuit the two coils of an inductometer, connected in series, 
the inductance of the whole can be varied in a known manner by 
rotating the smaller coil. On the other hand the introduction, or 
removal, of resistance without alteration of inductance cannot well 
be carried out with rigour. But in most cases the object can be 
sufficiently attained with the aid of a resistance-slide of thin 
German-silver wire which may be in the form of a nearly close 
loop. 

In the Wheatstone*s quadrilateral, as arranged for these ex- 
periments, the adjacent sides -R, S may be made of similar wires 
of German silver of equal resistance (^ ohm). If doubled they 
give rise to little induction, but the accuracy of the method is 
independent of this circumstance. The side P includes the 
conductor, or combination of conductors, under examination, an 
inductometer, and the resistance-slide. The other side, Q, must 
possess resistance and inductance greater than any of the con- 
ductors to be compared, but need not be susceptible of ready and 
measurable variations. In order to avoid mutual induction be- 
tween the branches, P and Q should be placed at some distance 
away, being connected with the rest of the apparatus by leads of 
doubled wire. 

It will be evident that when the interrupter acts in the 
battery branch, balance can be obtained at the telephone in the 
bridge only under the conditions that both the inductance and 
the resistance in P are equal in the aggi'egate to the correspond- 
ing quantities in Q. Hence when one conductor is substituted for 
another in P, the alterations demanded at the inductometer and 
in the slide give respectively the changes of inductance and of 
resistance. In this arrangement inductance and resistance are 
well separated, so that the results can be interpreted without 
calculation; but the movable contacts of the slide appear to 
introduce uncertainty into the determination of resistance. 

In order to get rid of the objectionable movable contacts 
some sacrifice of theoretical simplicity seems unavoidable. We 



235 ^] SYMMETRICAL ARRANGEMENT. 459 

can no longer keep the total resistances P and Q constant ; but by 
reverting to the arrangement adopted in a well-known form of 
Wheatstone's bridge, we cause the resistances taken from P to be 
added to Q, and vice versa. The transferable resistance is that of 
a straight wire of German-silver, with which one telephone ter- 
minal makes contact at a point whose position is read off on a 
divided scale. Any uncertainty in the resistance of this contact 
does not influence the measurements. 

Fig. 53 c. 




The diagram Fig. (53 c) shows the connection of the parts. One 
of the telephone terminals Tgoes to the junction of the {\ ohm) 
resistances R and S, the other to a point upon the divided vdre. 
The branch P includes one inductometer (with coils connected in 
series), the subject of examination, and part of the divided wire. 
The branch Q includes a second inductometer (replaceable by a 
simple coil possessing suitable inductance), a rheostat, or any 
resistance roughly adjustable from time to time, and the re- 
mainder of the divided wire. The battery branch By in which may 
also be included the interrupter, has its terminals connected, one 
to the junction of P and R, the other to the junction of Q and 8, 
When it is desired to use steady currents, the telephone can of 
course be replaced by a galvanometer. 

In this arrangement, as in the other, balance requires that the 
branches P and Q be similar in respect both of inductance and of 
resistance. The changes in inductance due to a shift in the 
movable contact may usually be disregarded, and thus any alte- 
ration in the subject (included in P) is measured by the rotation 
necessitated at the inductometer. As for the resistance, it is 
evident that (R and 8 being equal) the value for any additional 
conductor interposed in P is measured by twice the displacement 
of the sliding contact necessary to regain the balance. 

Experimental details of the application of this method to the 



460 ELECTRICAL VIBRATIONS. [235 L 

measurement of various combinations will be found in the paper* 
from which the above sketch is derived. Among these may be 
mentioned the verification of Maxwell's formulae, (8), (9) § 235 A', 
as to the influence of a neighbouring circuit, especially in the 
extreme case where the equivalent inductance is almost destroyed, 
and of the formula (10) § 235 vi relating to the behaviour of an 
electro-magnet shunted by a relatively high simple i-esistance. 
But the most interesting in many respects is the application to 
the phenomena presently to be considered, where the conductors 
in question are no longer approximately linear but must be 
regarded as solid masses in which the currents are distributed in 
a manner that needs to be determined by general electrical 
theoiy. 

As has already been remarked more than once, a leyden may 
always be supposed to be included in the circuit, the stiffness 
thereof having the effect of a negative inductance. If there be no 
hysteresis in the action of the leyden, the whole effect is thus 
represented ; but when the dielectric employed is solid, it appears 
that dissipative loss cannot be avoided. The latter effect manifests 
itself as an augmentation of apparent resistance, indistinguishable, 
unless the frequency be varied, from the ordinary resistance of the 
leads. A similar treatment may be applied to an electrolytic cell, 
the stiffness and resistance being presumably both functions of the 
frequency. 

236 u. It was proved by Maxwell* that a perfectly con- 
ducting sheet, forming a closed or an infinite surface, acts as a 
magnetic screen, no magnetic actions which may take place on 
one side of the sheet producing any magnetic effect on the other 
side. " In practice we cannot use a sheet of perfect conductivity ; 
but the above described state of things may be approximated to 
in the case of periodic magnetic changes, if the time-constants of 
the sheet circuits be large in comparison with the periods of t'he 
changes." 

"The experiment is made by connecting up into a primary 
circuit a battery, a microphone-clock, and a coil of insulated wire. 
The secondary circuit includes a parallel coil and a telephone. 
Under these circumstances the hissing sound is heard almost as 
well as if the telephone were inserted in the primary circuit 

* Phil. Mag.f loc, cit, 

2 Electricity and Magnititm, 1S78, § 655. 



235 t^.] ELECTROMAGNETIC SCREEN. 461 

itself. But if a large and stout plate of copper be interposed 
between the two coils, the sound is greatly enfeebled. By a proper 
choice of battery and of the distance between the coils, it is not 
difficult so to adjust the strength that the sound is conspicuous in 
the one case and inaudible in the other "^ 

One of the simplest applications of Maxwell's principle is to 
the case of a long cylindrical shell placed within a coaxal magnet- 
izing helix. The condition of minimum energy requires that such 
currents be developed in the shell as shall neutralize at internal 
points the action of the coil. Thus, if the conductivity of the 
shell be sufficiently high, the interior space is screened from the 
magnetizing force of periodic currents flowing in the outer helix, 
and conducting circuits situated within the shell must be devoid 
of induced currents. An obvious deduction is that the currents 
induced in a solid conducting core will be more and more confined 
to the neighbourhood of the surface as the frequency of electrical 
vibration is increased. 

The point at which the concentration of current towards the 
surface becomes important depends upon the relative values of the 
imposed vibration-period and the principal time-constant of the 
core circuit. If p be the specific resistance of the material, fi its 
magnetic permeability, a the radius of the cylinder, the expression 
for the induction (c) parallel to the axis, during the progress of the 
subsidence of free currents in a normal mode, is 

c^€^^J,{kr) (1). 

where k^^^^^!^ (2), 

P 

and ka is determined by the condition that 

J,{ka)^0 (3). 

The roots of (3) are, § 206, 

2-404, 5-520, 8654, 11-792, &c., 
so that for the principal mode of greatest persistence 

c = e^^ Jo (2-404 r/a) (4), 

where ^^J^^^Ip. (5). 

Acoustical Observations, PhiU Mag.^ vol. 13, p. 344, 1SS2. 



462 ELECTTRICAL VIBRATIONS. [235 W. 

For copper in c.o.s. measure p = 1642, /a= 1, and thus 



a» 



T = (-X)- = gOQnearly\ 

In the case of iron we may take as approximate values, /n = 100, 
^ = 10*. Thus for an iron wire of diameter (2a) equal to '33 cm., 
the value of t is about ^^ of a second, and is therefore comparable 
with the periods concerned in telephonic experiments. 

Regarded from an analytical point of view the theory of forced 
vibrations in a conducting core is equally simple, and was worked 
out almost simultaneously by Lamb', Oberbeck' and Heaviside*. 
In this case we are to regard X as given, equal (say) to tp, where 
pl^tr is the frequency. If I&^*^ be the imposed magnetizing force, 
the solution is 

"7^)'-""' <«• 

the value of k being given by (2). 

" When the period in the field is long in comparison with the 
time of decay of free currents, we have Jo(i'^)= 1, nearly, so that 
c is approximately constant and =/i/ throughout the section of 
the cylinder. But, in the opposite extreme, when the oscillations 
in the intensity of the field are rapid in comparison with the decay 
of free currents, the induced currents extend only to a small depth 
beneath the surface of the cylinder, the inner strata (so to speak) 
being almost completely sheltered firom electromotive force by the 
outer ones. Writing t^ = (1 — ifif, where 

27rAt^ 

we have, when qr is large, 

Jo (At) = const, x — '- — , 

approximately, and thence 

c = /i/ . 'sj{alr) . e^ <»-«) +'« t'-^* [e'^\ 
This indicates that the electrical disturbance in the cylinder 

1 ''On the Duration of Free Electric Currents in an Infinite Condncting 
Cylinder," Brit, Aksoc, Report for 1882, p. 446. 

2 Proc, Math. Soc, vol. xv., p. 189, Jan. 1884. 
» Wied, Ann., vol. xxi., p. 672, Ap. 1884. 

* EUctrieian^ May, 1884. Electrical Papers, vol. n., p. 868. 



(7), 



235 W.] CONDUCTING CORE. 463 

consists in a series of waves propagated inwards with rapidly 
diminishing amplitude*.** 

For experimental purposes what we most require to know is 

the reaction of the core currents upon the helix, in which alone 

we can directly measure electrical effects. This problem is fully 

treated by Heaviside*, but we must confine ourselves here to a 

mere statement of results. These are most conveniently expressed 

by the changes of effective inductance L and resistance R due to 

the core. If m be the number of turns per unit length in the 

magnetizing helix, and if hL, BR be the apparent alterations of L 

and R due to the introduction of the core, also reckoned per unit 

length, we have 

8Z = 4m-7r«a»(/iP-l)' 

SR^^m^TT'a^fjL.pQ J 

where P and Q are defined by 

P'iQ=^<t>'l<t> (8), 

the function <^ being of the form 

<^(a:) = /o(2iV^) = l+^+^^i-... + ^,^f" ^, -f (9), 

and the argument x being 

ipfi.ira^lp (10). 

If the material composing the core be non-conducting, a: = 0, and 

therefore 

P = l, Q = 0. 

Accordingly SL = 4mVa« (m - 1), Si2 = (11). 

These values apply also, whatever be the conductivity of the 
core, if the frequency be suflSciently low. 

At the other extreme, when jp = oo , we require the ultimate 
form of 07<^. From the value of Jq given in (10) § 200, or other- 
wise, it may be shewn that in the limit 

<t>'l4>^x-i (12), 

so that P = Q=-— — 1 — _. (13). 

The introduction of these values into (7) shews that in the 
limit, when the frequency is exceedingly high, 

SX«-47nVaS Si2 = (14), 

^ Lamb, loc, cit.y where is also disoassed the problem of the currents induced by 
the sudden cessation of a previously constant field. 
* loc, ciU 



464 ELECTRICAL VIBRATIONS. [235 U. 

as might also have been inferred from the consideration that the 
induced currents are then confined to the sur&ce of the core. 

An example of the application of these formulae to an inter- 
mediate case and a comparison with experiment will be found in 
the paper already referred to^ 

236 v. The application of Maxwell's principle to the case of 
a wire, in which a longitudinal electric current is induced, is less 
obvious; and Heaviside' appears to have been the first to state 
distinctly that the current is to be regarded as propagated inwards 
from the exterior. The relation between the electromotive force 
E and the total current C had, however, been given many years 
earlier by Maxwell' in the form of a series. His result is equi- 
valent to 

^^^tpl/R.A^^-^^~jj^^ (1), 

in which R denotes the whole resistance of the length I to 
steady currents, /i the permeability, and />/27r the frequency. The 
function </> is that defined by (9) § 235 u, and il is a constant 
dependent upon the situation of the return current* 

The most convenient form of the results is that which we have 
already several times employed. If we write 

E^RC^ipLV (2), 

in which R' and L' are real, these quantities will represent the 
effective resistance and inductance of the wire. When the argu- 
ment in (1) is small, that is when the frequency is relatively low, 
we thus obtain 

i7i = 4+M{i-A^'|,^'+3Ht;^* + ...} (4)». 

1 Phil. Mag., vol. 22, p. 493, 1886. 

'^ Electrician, Jan., 1885 ; Electrical Papers, vol. i., p. 440. 

=» Phil. Trans., 1865 ; Electricity and Magnetism, vol. ii., § 690. 

* The simplest case arises when the dielectric, which bounds the cylindrical 
wire of radius a, is enclosed within a second conducting mass extending outwards 
to infinity and bounded internally at a cylindrical surface r=6. We then have 
^ =2 log (6/a). See J. J. Thomson, loc. cit., § 272. 

' Phil. Mag., vol. 21, p. 387, 1886. It is singular that Maxwell (loc. cit.) seems 
to have regarded his solution as conveying a correction to the self-induction only of 
the wire. 



235 v.] LIMITING FORMS. 465 

When p is very small, these equations give, as was to be 
expected, 

R^R, i'=:i(ii+i/i) (5). 

If we include the next terms, we recognise that, in accordance 
with the general rule, L begins to diminish and R to increase. 

When p is very great, we have to make use of the limiting 
form of 4>I4>' As in § 235 u, 

<^/</>' = (l+i)VapWii) (6); 

and thus ultimately 

R=^s/(hplf^R) (7), 

r/l=:A'hy/(fiR/2pl) (8), 

the first of which increases without limit with p, while the second 
tends to the finite limit A, corresponding to the total exclusion of 
current from the interior of the wire. 

Experiments^ upon an iron wire about 18 metres long and 8*3 
millimetres in diameter led to the conclusion that the resistance 
to variable currents of frequency 1050 was such that R/R = 1*9. 
A calculation based upon (1) shewed that this result is in harmony 
with theory, if /a = 99'5. Such is about the value indicated by 
other telephonic experiments. 

236 w. The theory of electric currents in such wires as are 
commonly employed in laboratory experiments is simple, mainly in 
consequence of the subordination of electrostatic capacity. When 
this element can be neglected, the current is necessarily the same 
at all points along the length of the wire, so that whatever enters 
a wire at the sending end leaves it unimpaired at the receiving 
end. In this case the whole electrical character of the wire can 
be expressed by two quantities, its resistance R and inductance L, 
and these may usually be treated as constants, independent of the 
frequency. The relation of the current to the electromotive force 
under such circumstances has already been discussed (7) § 235 j. 
When we have occasion to consider only the amplitude of the 
current, irrespective of phase, we may regard it as determined 
by V[^+l>*^']> a quantity which is called by Heaviside the 
impedance. Thus in circuits devoid of capacity the impedance is 
always increased by the existence of L, 

1 Phil. Mag,, yol. 22, p. 4S8, 1886. 

R. 30 



466 ELECTRICAL VIBRATIONS. [235 W. 

Circuits employed for practical telephony may often be re- 
garded as coming under the above description, especially when 
the wires are suspended and are of but moderate length. But 
there are other cases in which electrostatic capacity is the domi- 
nating feature. The theory of electric cables was established 
many years ago by Lord Kelvin^ for telegraphic purposes. If 8 
be the capacity and R the resistance of the cable, reckoned per 
unit length, V and C the potential and the current at the point z, 

we have 

SdVldt=^-dCldz, RC=^dVldz (1), 

whence R8dC/dt^d^C/dz^ (2), 

the well known equation for the conduction of heat discussed by 

Fourier, On the assumption that C is proportional to c*^, it 

reduces to 

d»C/d^2={V(ipiifif).(l+i)}»C (3); 

so that the solution for waves propagated in the positive direc- 
tion is 

C=Coe-"^^^p^^"cos{pt'-y/(^pRS).z} (4). 

The distance in traversing which the current is attenuated in the 

ratio of e to 1 is thus 

z = ^(2/pRS) (5). 

A very slight consideration of the magnitudes involved is 
suflScient to give an idea of the diflBculty of telephoning through a 
long cable. If, for example, the frequency {pj^ir) be that of a 
note rather more than an octave above middle c, and the cable be 
such as are used to cross the Atlantic, we have in C.G.S. measure 

Vi> = 60, (iJS)-i = 2 X 10", 

and accordingly from (5) 

2: = 3 X 10* cm. = 20 miles approximately. 

A distance of 20 miles would thus reduce the intensity of 
sound, measured by the square of the amplitude, to about a 
tenth, an operation which could not be repeated often vdthout 
rendering it inaudible. With such a cable the practical limit 
would not be likely to exceed fifty miles, more especially as 
the easy intelligibility of speech requires the presence of tones 
still higher than is supposed in the above numerical exam pie ^ 

* Proc, Roy, Soc.y 1855 ; Mathematical and Phytical Papers^ vol. u. p. 61. 
3 '" On Telephoning through a Cable.*' Brit. An, Report for 1884, p. 632. 



235 x.'\ heaviside's theory. 467 

236 X, In the above theory the insulation is supposed to be 
perfect and the inductance to be negligible. It is probable that 
these conditions are suflSciently satisfied in the case of a cable, 
but in other telephonic lines the inductance is a feature of great 
importance. The problem has been treated with full generality 
by Heaviside, but a slight sketch of his investigation is all that 
our limits permit. 

If R, S, L, K he the resistance, capacity or permittance, in- 
ductance, and leakage-conductance respectively per unit of length, 
V and C the potential-difference and current at distance z, the 
equations, analogous to (1) § 235 ti;, are 

Thus, if the currents are harmonic, proportional to e*^S 

'^^^(R + ipL){K + ipS)C. (2). 

with a similar equation for F. 

It might perhaps have been expected that a finite leakage K 
would always act as a complication; but Heaviside^ has shewn 
that it may be so adjusted as to simplify the matter. This case, 
which is remarkable in itself and also serves to throw light upon 
the general question, arises when R/L = K/S. We will write 

ififtr»=l, R/L=^K/8 = q (2), 

where i; is a velocity of the order of the velocity of light. The 
equation for V is then by (1) 

i^d'V/dz^=:(d/dt-{-qyV (3); 

or if we take U so that 

V^e-^U (4), 

v^cPU/dz' = d'U/dt^ (5), 

the well-known equation of undisturbed wave propagation § 144. 
"Thus, if the wave be positive, or travel in the direction of 
increasing z, we shall have, if /i (z) be the state of V initially, 

V,^e-^Mz-vt\ C,= V,ILv (6). 

If Fa, Cj be a negative wave, travelling the other way, 

V,^e'^Mz + vt\ Cr V,ILv (7). 

^ Electrician, June 17, 18S7. Electrical Papers, vol xi. pp. 126, 809. 



468 ELECTRICAL VIBRATIONS. [235 X, 

Thus, any initial state being the sum of F^ and Fj to make F, 
and of Cx and C^ to make (7, the decomposition of an arbitrarily 
given initial state of F and G into the waves is effected by 

V,^\{V^vLC\ y,^\{V^vLC) (8). 

We have now merely to move Fj bodily to the right at speed 
V, and F, bodily to the left at speed r, and attenuate them to the 
extent e""^, to obtain the state at time i later, provided no changes 
of condition have occurred. The solution is therefore true for all 
future time in an infinitely long circuit. But when the end of a 
circuit is reached, a reflected wave usually results, which must be 
added on to obtain the real result." 

As in § 144, the precise character of the reflection depends 
upon the terminal conditions. "One case is uniquely simple. 
Let there be a resistance inserted of amount vL. It introduces 
the condition V — vLG if at say fi, the positive end of the circuit, 
and V^—vLC if at the negative end, or beginning. These are 
the characteristics of a positive and of a negative wave respect- 
ively ; it follows that any disturbance arriving at the resistance is 
at once absorbed. Thus, if the circuit be given in any state 
whatever, without impressed force, it is wholly cleared of electrifi- 
cation and current in the time l/v at the most, if I be the length 
of the circuit, by the complete absorption of the two waves into 
which the initial state may be decomposed." 

" But let the resistance be of amount R^ at say B ; and let Fj 
and Fj be corresponding elements in the incident and reflected 
waves. Since we have 

F, = t;ZCx, V, — vLC,, F,+ F, = i2,((7, + C,)...(9), 

we have the reflected wave given by 

F, R,^vL 

rr-R^TVL (10)- 

If Ri be greater than the critical resistance of complete ab- 
sorption, the current is negatived by reflection, whilst the electri- 
fication does not change sign. If it be less, the electrification is 
negatived, whilst the current does not reverse." 

"Two cases are specially notable. They are those in which 
there is no absorption of energy. If ^=0, meaning a short 
circuit, the reflected wave of F is a perverted and inverted copy of 



235 a:.] heaviside's theory. 469 

the incident. But if iJ = « , representing insulation, it is G that 
is inverted and perverted \" 

The cases last mentioned are evidently analogous to the reflec- 
tion of a sonorous aerial wave travelling in a pipe. If the end of 
the pipe be closed, the reflection is of one character, and if it be 
open of another character. In both cases the whole energy is 
reflected, § 257. The waves reflected at the two ends of an electric 
circuit complicate the general solution, especially when the sim- 
plifying condition (2) does not hold. But in many cases of 
practical interest they may be omitted without much loss of 
accuracy. One passage over a long line usually introduces con- 
siderable attenuation, and then the effect of the reflected wave, 
which must traverse the line three times in all, becomes insigni- 
ficant. 

In proceeding to the general solution of (2) for a positive 
wave, we will introduce, after Heaviside, the following abbrevia- 
tions, 

n'LS^l. R/Lp=/. K/Sp = g (11). 

In terms of these quantities (2) may be written 

cPC/dz^^iP-^-iQyC (12), 

where 

P« or Q» = J {p/vY {(1 +/»)* (1 +f)i ± (fg - 1)} ... (13). 

Thus, if P and Q be taken positively, the solution for a wave 
travelling in the positive direction is 

C-=^Coe'P'cos{pt'Qz) (14), 

the current at the origin being Cq cos pt 

The cable formula, § 235 w, is the particular case arrived at by 
supposing in (13)/= oo , ^r = 0, which then reduces to 

p^^Q^^^pRS (15). 

Again, the special case of equation (3) is derivable by putting 
/ = g = q/p' The result is 

P = qlv, Q=p/v (16). 

If the insulation be perfect, g = 0, and (13) becomes 

P« or (? = i(p/^)'{(l +/')*?!} (17). 

^ Heayiside, Collected Works^ vol. n. p. 312. 



470 ELECTRICAL VIBRATIONS. [235 X. 

In certain examples of long copper lines of high conductivity, 
/ may be regarded as small so far as telephonic frequencies are 
concerned. Equation (17) then gives 

P=:pf/2v^R/2vL, Q^pjv (18). 

For a further discussion of the various cases that may arise 
the reader must be referred to the writings of Heaviside already 
cited. The object is to secure, as far as may be, the propagation 
of waves without alteration of type. And here it is desirable to 
distinguish between simple attenuation and distortion. If, as in 
(16) and (18), P is independent of p, the amplitudes of all com- 
ponents are reduced in the same ratio, and thus a complex wave 
travels without distortion. The cable formula (15) is an example 
of the opposite state of things, where waves of high frequency are 
attenuated out of proportion to waves of low frequency. It appears 
from Heaviside's calculations that the distortion is lessened by 
even a moderate inductance. 

The eflFectiveness of the line requires that neither the attenua- 
tion nor the distortion exceed certain limits, which however it is 
hard to lay down precisely. A considerable amount of distortion 
is consistent with the intelligibility of speech, much that is 
imperfectly rendered being supplied by the imagination of the 
hearer. 

236 y. It remains to consider the transmitting and receiving 
appliances. In the early days of telephony, as rendered practical 
by Graham Bell, similar instruments were employed for both 
purposes. Bell's telephone consists of a bar magnet, or battery 
of bar magnets, provided at one end with a short pole-piece 
which serves as the core of a coil of fine insulated wire. In close 
proximity to the outer end of the pole-piece is placed a circular 
disc of thin iron, held at the circumference. Under the influence 
of the permanent magnet the disc is magnetized radially, the 
polarity at the centre being of course opposite to that of the 
neighbouring end of the steel magnet. 

The operation of the instrument as a transmitter is readily 
traced. When sonorous waves impinge upon the disc, it responds 
with a symmetrical transverse vibration by which its distance 
from the pole-piece is alternately increased and diminished. 
When the interval is diminished, more induction passes through 
the pole-piece, and a corresponding electro-motive force acts in 



235 y.] bell's telephone. 471 

the enveloping coil. The periodic movement of the disc thus 
gives rise to a periodic current in any circuit connected with the 
telephone coil. 

The electro-motive force is in the first instance proportional 
to the permanent magnetism to which it is due; and this law 
would continue to hold, were the behaviour of the pole-piece and 
of the disc conformable to that of the " soft iron *' of approximate 
theory. But as the magnetism rises, and the state of saturation 
is more nearly approached, the response to periodic changes of 
force becomes feebler, and thus the efficiency falls below that 
indicated by the law of proportionality. If we could imagine the 
state of saturation in the pole-piece to be actually attained, the 
induction through the coil would become almost incapable of 
variation, being reduced to such as might occur were the iron 
removed. There is thus a point, dependent upon the properties 
of magnetic matter, beyond which it is pernicious to raise the 
amount of the permanent magnetism ; and this point marks the 
maximum efficiency of the transmitter. It is probable that the 
most favourable condition is not fully reached in instruments 
provided with steel magnets; but the considerations above 
advanced may serve to explain why an electro-magnet is not 
substituted. 

The action of the receiving instrument may be explained on 
the same principles. The periodic current in the coil alternately 
opposes and cooperates with the permanent magnet, and thus the 
iron disc is subjected to a periodic force acting at its centre. 
The vibrations are thence communicated to the air, and so reach 
the ear of the observer. As in the case of the transmitter, the 
efficiency attains a maximum when the magnetism of the pole- 
piece is still far short of saturation. 

The explanation of the receiver in terms of magnetic forces 
pulling at the disc is sometimes regarded as inadequate or even as 
altogether wide of the mark, the sound being attributed to " mole- 
cular disturbances " in the pole-piece and disc. There is indeed 
every reason to suppose that molecular movements accompany 
the change of magnetic state, but the question is how do these 
movements influence the ear. It would appear that they can do 
so only by causing a transverse motion of the surface of the disc, 
a motion from which nodal subdivisions are not excluded. 



472 ELECTRICAL VIBRATIONS, [235 y. 

In support of the " push and pull theory " it may be useful to 
cite an experiment tried upon a bipolar telephone. In this 
instrument each end of a horse-shoe magnet is provided with a 
pole-piece and coil, and the two pole-pieces are brought into 
proximity with the disc at places symmetrically situated with 
regard to the centre. In the normal use of the instrument the 
two coils are permanently connected as in an ordinary horse-shoe 
electro-magnet, but for the purposes of the experiment provision 
was made whereby one of the coils could be reversed at pleasure 
by means of a reversing key. The sensitiveness of the telephone 
in the two conditions was tested by including it in the circuit of 
a Daniell cell and a scraping contact apparatus, resistance from a 
box being added until the sound was but just easily audible. 
The resistances employed were such as to dominate the self- 
induction of the circuit, and the comparison shewed that the 
reversal of the coil from its normal connection lowered the sensi- 
tiveness to current in the ratio of 11 : 1. That the reduction was 
not still greater is readily explained by outstanding failures of 
symmetry; but on the "molecular disturbance" theory it is not 
evident why there should be any reduction at all. 

Dissatisfaction with the ordinary theory of the action of a 
receiving telephone may have arisen from the difficulty of under- 
standing how such very minute motions of the plate could be 
audible. This is, however, a question of the sensitiveness of the 
eai', which has been proved capable of appreciating an amplitude 
of less than 8 x 10~®cra.^ The subject of the audible minimum 
will be further considered in the second volume of this work. 

The calculation a priori of the minimum current that should 
be audible in the telephone is a matter of considerable difficulty ; 
and even the determination by direct experiment has led to 
widely discrepant numbers. In some recent experiments by the 
author a unipolar Bell telephone of 70 ohms resistance was 
employed. The circuit included also a resistance box and an 
induction coil of known construction, in which acted an electro- 
motive force capable of calculation. Up to a frequency of 307 
this could be obtained from a revolving magnet of known moment 
and situated at a measured distance from the induction coil. For 
the higher frequencies magnetized tuning-forks, vibrating with 
measured amplitudes, were substituted. In either case the 

^ Proc, Roy, Soc, vol. xxvi. p. 248, 1877. 



235 y.] 



MINIMUM CURRENT. 



473 



resistance of the^ circuit was increased until the residual sound 
was but just easily audible. Care having been taken so to 
arrange matters that the self-induction of the circuit was negli- 
gible, the current could then be deduced from the resistance and 
the calculated electro-motive force operating in the induction 
coil. The following are the results, in which it is to be under- 
stood that the currents recorded might have been halved without 
the sounds being altogether lost : 



Pitch 


Source 

Fork 
Revolving Magnet 

Fork 
Revolving Magnet 

Fork 


Current in 
10~® amperes 

2800 
250 
83 
49 
32 
15 
7 

4-4 
10 


128 

192 

256 . 

307 

320 

384 

512 

640 

768 











The effect of a given current depends, of course, upon the 
manner in which the telephone is wound. If the same space be 
occupied by the copper in the various cases, the current capable of 
producing a particular effect is inversely as the square root of the 
resistance. 

The numbers in the above table giving the results of the 
author's experiments are of the same order of magnitude as 
those found by Ferraris^ whose observations, however, related 
to sounds that were not pure tones. But much lower estimates 
have been put forward. Thus Tait' gives 2 x 10~" amperes, 
and Preece a still lower figure, 6 x 10~". These discrepancies, 
enormous as they stand, would be still further increased were 
the comparison made to refer to the amotmts of energy absorbed. 

According to the calculations of the author the above tabulated 
sensitiveness to a periodic current of frequency 256 is about what 
might reasonably be expected on the push and pull theory*. At 

1 Atti della Accad, d, Sci. Di Torino, vol. xiii. p. 1024, 1S77. 

2 Edin. Proc, vol. ix. p. 551, 187S. 

3 I propose shortly to publish these calculations. 



1 



474 ELECTRICAL VIBRATIONS. [235 y. 

this frequency, which is below those proper to the telephone plate 
(§ 221 a), the motion of the plate is governed by elasticity rather 
than by inertia, and an equilibrium theory (§ 100) is applicable as 
a rough approximation. The greater sensitiveness of the telephone 
at frequencies in the neighbourhood of 512 would appear to 
depend upon resonance (§ 46). It is doubtful whether the much 
higher sensitiveness claimed by Tait and Preece could be re- 
conciled with theory. 

It appears to be established that the iron plate of a telephone 
may be replaced by one of copper, or even of non-conducting 
material, without absolute loss of sound; but these effects are 
probably of a diflFerent order of magnitude. In the case of copper 
induced currents may confer the necessary magnetic properties. 
For a description of the ingenious receiver invented by Edison 
and for other information upon telephonic appliances the reader 
may consult Preece and Stubbs* Manual of Telephony. 

In existing practice the transmitting instrument depends 
upon a variable contact. The first carbon transmitter was con- 
structed by Edison in 1877, but the instruments now in use are 
modifications of Hughes' microphone \ A battery current is led 
into the line through pieces of metal or of carbon in loose juxta- 
position, carbon being almost universally employed in practice. 
Under the influence of sonorous vibration the electrical resistance 
of the contacts varies, and thus the current in the line is rendered 
representative of the sound to be reproduced at the receivihg 
end. 

That the resistance of the contact should vary with the 
pressure is not surprising. If two clean convex pieces of metal 
are forced together, the conductivity between them is represented 
by the diameter of the circle of contact (§306). The relation 
between the circle of contact and the pressure with which the 
masses are forced together has been investigated in detail by 
Hertz ^ His conclusion for the case of two equal spheres is that 
the cube of the radius of the circle of contact is proportional to 
the pressure and to the radii of the spheres. But it has not yet 
been shewn that the action of the microphone can be adequately 
explained upon this principle. 

1 Proc. Roy. Soc, vol. xxvii. p. 862, 1878. 
' Crelle, Jonm. Math. xcn. p. 156, 1882. 



APPENDIX. 



ON PROGRESSIVE WAVES. 

From the Proceedings of the London Mathematical Societyy 

Vol. IX., p. 21, 1877. 

It has often been remarked that, when a group of waves advances 
into still water, the velocity of the group is less than that of the indi- 
vidual waves of which it is composed ; the waves appear to advance 
through the group, dying away as they approach its anterior limit. 
This phenomenon was, I believe, lirst explained by Stokes, who re- 
garded the group as formed by tlie superposition of two infinite trains 
of waves, of equal amplitudes and of nearly equal wave-lengths, ad- 
vancing in the same direction. My attention was called to the subject 
about two years since by Mr Froude, and the same explanation then 
occurred to me independently*. In my book on the "Theory of 
Sound" (§191), I have considered the question more generally, and 
have shewn that, if V be the velocity of propagation of any kind of 
waves whose wave-length is X, and k = 27r/X, then U, the velocity of 
a group composed of a great number of waves, and moving into an un- 
disturbed part of the medium, is expressed by 



^-'4P (.), 



dk 

* Another phenomenon, also mentioned to me by Mr Froude, admits of a similar 
explanation. A steam-launch moving quickly through the water is accompanied by 
a peculiar system of diverging waves, of which the most striking feature is the 
obliquity of the line containing the greatest elevations of successive waves to the 
wave-fronts. This wave pattern may be explained by the superposition of two (or 
more) infinite trains of waves, of slightly dififering wave-lengths, whose directions 
and velocities of propagation are so related in each case that there is no change of 
position relatively to the boat. The mode of composition will be best understood by 
drawing on paper two sets of parallel and equidistant lines, subject to the above 
condition, to represent the crests of the component trains. In the case of two trains 
of slightly different wave-lengths, it may be proved that the tangent of the angle 
between the line of maxima and the wave-fronts is half the tangent of the angle 
between the wave-fronts and the boat's course. 



476 APPENDIX. 

or, as we may also write it, 

''^''-'*'-^' » 

Thus, if rxX% U={l-'n) V (3). 

In fact, if the two infinite trains be represented by cos k{Vt'- x) 
and cos k' ( V't — x), their resultant is represented by 

cos k{Vt''X) + cos k' ( V't - x), 
which is equal to 

;k'V'-kV k'^k 1 (k'V'-k-kV A' + Jfc 



2 cos 



(k'V'-kV k'^k ) (k'V''¥kV k-^k 1 



If k' -k, V - F be small, we have a train of waves whose amplitude 
varies slowly from one point to another between the limits and 2, 
forming a series of groups separated from one another by regions com- 
paratively free from disturbance. The position at time t of the middle 
of that group, which was initially at the origin, is given by 

{k'r^kV)t^{k'^k)x=0, 

which shews that the velocity of the group is {k' F' - A; F) -f {k* — k). 
In the limit, when the number of waves in each group is indefinitely 
great, this result coincides with (1). 

The following particular cases are worth notice, and are here tabu- 
lated for convenience of comparison : — 

F oc X, U=0, Reynolds* disconnected pendulums. 

F X X*, U = ^ F, Deep-water gravity waves. 

V X X®, U = F, Aerial waves, «fec. 

F X X"i, U = ^V, Capillary water waves. 

F X X"^ U=2Vy Flexural waves. 

The capillary water waves are those whose wave-length is so small 
that the force of restitution due to capillarity largely exceeds that due 
to gravity. Their theory has been given by Thomson (PhiL Mag., 
Nov. 1871). The fiexural waves, for which U=2Vy are those cor- 
responding to the bending of an elastic rod or plate ("Theory of 
Sound," g 191). 

In a paper read at the Plymouth meeting of the British Association 
(afterwards printed in "Nature," Aug. 23, 1877), Prof. Osborne 
Reynolds gave a dynamical explanation of the fact that a group of 
deep-water waves advances with only half the rapidity of the indi- 
vidual waves. It appears that the energy propagated across any point, 
when a train of waves is passing, is only one-half of the energy neces- 



PROGRESSIVE WAVES. 477 

saiy to supply the waves which pass in the same time, so that, if the 
train of waves be limited, it is impossible that its front can be propa- 
gated with the full velocity of the waves, because this would imply the 
acquisition of more energy than can in fact be supplied. Prof. Reynolds 
did not contemplate the cases where more energy is propagated than 
corresponds to the waves passing in the same time ; but his argument, 
applied conversely to the results already given, shews that such cases 
must exist. The ratio of the energy propagated to that of the passing 
waves is U '. V; thus the energy propagated in the unit time is U : V 
of that existing in a length F, or U times that existing in the unit 
length. Accordingly 

Energy propagated in unit time : Energy contained (on an average) 
in unit length =d{kV) : dk, by (1). 

As an example, I will take the case of small irrotational waves in 
water of finite depth ^. If 2; be measured downwards from the surface, 
and the elevation (h) of the wave be denoted by 

h = 11 cos (nt — kx) (4), 

in which n = kVj the corresponding velocity-potential (^) is 

^ = - VII — j^ -j^i — sin (nt - kx) (5). 

This value of <f> satisfies the general differential equation for irrota- 
tional motion (v*^ = 0), makes the vertical velocity dift/dz zero when 
z = l, and — cUi/dt when z = 0. The velocity of propagation is given by 

^ -kl^T6^ ^^^• 

We may now calculate the energy contained in a length ac, which is 
supposed to include so great a number of waves that fractional parts 
may be left out of account. 

For the potential energy we have 

V^ = gpJlzdzdx = yplh^dx = ypIP.x (7). 

For the kinetic energy, 

=yj{*'^)^^'^=yp^''^ W' 

by (1) and (6). If, in accordance with the argument advanced at the 

* Prof. Reynolds considers the trochoidal wave of Rankine and Fronde, which 
involves molecular rotation. 



478 APPENDIX. 

end of this paper, the equality of Vi and T be assumed, the value of 
the velocity of propagation follows from the present expressions. The 
whole energy in the waves occupying a length x is therefore (for each 

unit of breadth) V^ -h T = igpll^ , x (9), 

// denoting the maximum elevation. 

We have next to calculate the energy propagated in time t across a 
plane for which x is constant, or, in other words, the work ( W) that 
must be done in order to sustain the motion of the plane (considered 
as a flexible lamina) in the face of the fluid pressures acting upon the 
front of it. The variable part of the pressure (Sp), at depth «, is 
given by 

^/^ = - P -57 = - ^ ^^ — ^rr^— cos (^« - ^) > 
while for the horizontal velocity 

^ = ^^^ e*^-e-^ cos(n<-Aa;); 

so that W^jJBp^dzdt^igpHKVt.^l-^^^i^^'^ (10), 

on integration. From the value of F in (6) it may be proved that 

and it is thus verified that the value of W for a unit time 

= -\i — ^ energy in umt length. 

As an example of the direct calculation of U, we may take the case 
of waves moving under the joint influence of gravity and cohesion. 

It is proved by Thomson that 

V^ = l^T'k, (11), 

where T' is the cohesive tension. Hence 

When k is small, the surface tension is negligible, and then Vt^^V; 
but when, on the contrary, k is large, U=^V, as has already been 
stated. When TJ^ = g, U = V, This corresponds to the miniin iiTn 
velocity of propagation investigated by Thomson. 



PROGRESSIVE WAVES. 479 

Although the argument from interference groups seems satisfactory, 
an independent investigation is desirable of the relation between 
energy existing and energy propagated. For some time I was at a loss 
for a method applicable to all kinds of waves, not seeing in particular 
why the comparison of energies should introduce the consideration of 
a variation of wave-length. The following investigation, in which the 
increment of wave-length is imaginary, may perhaps be considered to 
meet the want : — 

Let us suppose that the motion of every part of the medium is 
resisted by a force of very small magnitude proportional to the mass 
and to the velocity of the part, the effect of which will be that waves 
generated at the origin gradually die away as x increases. The motion, 
which in the absence of friction would be represented by cos {nt — kx), 
under the influence of friction is represented by c"'^ cos (nt — kx), 
where /i is a small positive coefficient. In strictness the value of k is 
also altered by the friction; but the alteration is of the second order as 
regards the frictional forces, and may be omitted under the circum- 
stances here supposed. The energy of the waves per unit length at 
any stage of degradation is proportional to the square of the amplitude, 
and thus the whole energy on the positive side of the origin is to the 
energy of so much of the waves at their greatest value, t.«., at the 

origin, as would be contained in the unit of length, as J^ e~^' dx : I, 

or as (2/a)"^ : 1. The energy transmitted through the origin in the 
unit time is the same as the energy dissipated ; and, if the frictional 
force acting on the element of mass m be Amv, where v is the velocity 
of the element and h is constant, the energy dissipated in unit time is 
Ji^in^ or 2hT, T being the kinetic energy. Thus, on the assumption 
that the kinetic energy is half the whole energy, we find that the 
energy transmitted in the unit time is to the greatest energy existing 
in the unit length as h : 2/i. It remains to find the connection be- 
tween h and fi. 

For this purpose it will be convenient to regard cos {nt - kx) as the 
real part of e^^ e***, and to inquire how k is affected, when n is given, 
by the introduction of friction. Now the effect of friction is repre- 
sented in the differential equations of motion by the substitution of 
d^jdi^ + hdjdt in place of d^/dfi, or, since the whole motion is proportional 
to e^^, by substituting — ?i* + ihn for — n*. Hence the introduction of 
friction corresponds to an alteration of n from n to n — ^i/i (the square 
of h being neglected); and accordingly k is altered from k to 
k — ^Vidkjdn, The solution thus becomes g -i* «<**/<«» ^(»<-*»)^ or, when 
the imaginary part is rejected, g-i**<'*/<'» cos {nt - kx) ; so that 
/A = ^A dkjdn, and h : 2fi = dn/dk. The ratio of the energy transmitted 



480 APPENDIX. 

in the unit time to the energy existing in the unit length is therefore 
expressed by dnjdk or d {k V)ldk, as was to be proved 

It has often been noticed, in particular cases of progressive waves, 
that the potential and kinetic energies are equal ; but I do not call to 
mind any general treatment of the question. The theorem is not 
usually true for the individual parts of the medium*, but must be 
imderstood to refer either to an integral number of wave-lengths, or to 
a space so considerable that the outstanding fractional parts of waves 
may be left out of account. As an example well adapted to give in- 
sight into the question, I will take the case of a uniform stretched 
circular membrane (" Theory of Sound," § 200) vibrating with a given 
niunber of nodal circles and diameters. The fundamental modes are 
not quite determinate in consequence of the symmetry, for any dia- 
meter may be made nodal. In order to get rid of this indeterminate- 
ness, we may suppose the membrane to carry a small load attached to 
it anywhere except on a nodal circle. There are then two definite 
fundamental modes, in one of which the load lies on a nodal diameter, 
thus producing no e£fect, and in the other midway between nodal dia- 
meters, where it produces a maximum effect ("Theory of Sound," 
J5 208). If vibrations of both modes are going on simultaneously, the 
potential and kinetic energies of the whole motion may be calculated 
by simple additicyn of those of the components. Let us now, supposing 
the load to diminish without limit, imagine that the vibrations are of 
equal amplitude and differ in phase by a quarter of a period. The 
result is a progressive wave, whose potential and kinetic energies are 
the sums of those of the stationary waves of which it is composed. 
For the first component we have 7^ = E cos' nt, T^ = E sin' ivt ; and 
for the second component, V^^ E sin' n/, T^=: E cos' lU ; so that 
]\+ V.i = Ti+ T2- Ey or the potential and kinetic energies of the 
progressive wave are equal, being the same as the whole energy of 
either of the components. The method of proof here employed appears 
to be sufficiently general, though it is rather difficult to express it in 
language which is appropriate to all kinds of waves. 

* Atrial waves are an important exception. 



END OF VOL. I. 



CAMBBIDOE : PRINTED BT C. J. CLAT, M.A. AMD SONS, AT THB UNIYEB8ITT PBB88.